An Introduction to Derivative Securities, Financial Markets, and Risk Management: 2nd Edition [2 ed.] 1944659552, 9781944659554

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Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management ROBERT JARROW

Second Edition

ARKADEV CHATTERJEA

World Scientific

AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management Second Edition

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AN INTRODUCTION TO

Derivative Securities, Financial Markets, and Risk Management Second Edition

ROBERT A. JARROW Cornell University

ARKADEV CHATTERJEA Indiana University Bloomington

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TAIPEI

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CHENNAI

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TOKYO

Published by World Scientific Publishing Co. Inc. 27 Warren Street, Suite 401-402, Hackensack, NJ 07601, USA Head office: 5 Toh Tuck Link, Singapore 596224 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Jarrow, Robert A., author. | Chatterjea, Arkadev, author. Title: An introduction to derivative securities, financial markets, and risk management / Robert Jarrow (Cornell University), Arkadev Chatterjea (Indiana University). Description: 2nd edition. | New Jersey : World Scientific, [2019] | Includes bibliographical references and index. Identifiers: LCCN 2018022534 | ISBN 9781944659554 Subjects: LCSH: Derivative securities. | Financial institutions. | Capital market. | Risk management. Classification: LCC HG6024.A3 J3747 2018 | DDC 332.64/57--dc23 LC record available at https://lccn.loc.gov/2018022534

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2019 by Robert Jarrow and Arkadev Chatterjea All rights reserved.

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Y0018#t=suppl

Desk Editors: Karimah Samsudin and Shreya Gopi

Typeset by diacriTech

Printed in Singapore

Bob: To my wife Gail for her patience and understanding. Arka: To my wife Sudeshna for her cheerful and steadfast support, and to our daughters Rushtri, Tvisha, and Roudra (all younger than the book!), who also cheerfully and proudly supported my writing.

About the Authors Robert A. Jarrow is the Ronald P. and Susan E. Lynch Professor of Investment Management at the Samuel Curtis Johnson Graduate School of Management, Cornell SC Johnson College of Business. He is among the most distinguished finance scholars of his generation. Jarrow has done research in nearly all areas of derivatives pricing. He is the co-developer of two widely used pricing models in finance, the Heath–Jarrow– Morton (HJM) model for pricing interest-rate derivatives and the reduced form model for pricing securities with credit risk. He is the author of more than 200 academic publications, seven books including Option Pricing (with Andrew Rudd, 1983), Modelling Fixed Income Securities and Interest Rate Options (1996), and Derivative Securities (with Stuart Turnbull, 2000), and several edited volumes. Arkadev Chatterjea is a Visiting Professor of Finance, Kelley School of Business, Indiana University Bloomington. He is also a Research Fellow at UNC Chapel Hill and a Visiting Fellow at CHERI, Cornell University. He did his Ph.D. at Cornell, where he was a student of Jarrow. Earlier, he was a professor of finance at the Indian Institute of Management Calcutta. A winner of research and teaching awards in the USA, Chatterjea has taught derivatives at the above universities and at other institutions including CU Boulder, the Helsinki School, Hong Kong UST, and IIM Ahmedabad. Photo by Mr. Kallol Nath.

Brief Contents

About the Authors VI Preface to Second Edition XXVII Preface to First Edition XXXI

17 Single-Period Binomial Model 367 18 Multiperiod Binomial Model 392 19 The Black–Scholes–Merton Model 422

PART I Introduction 1 Derivatives and Risk Management 2 2 Interest Rates 22

20 Using the Black–Scholes–Merton Model 462

3 Stocks 53

PART IV Interest Rate Derivatives

4 Forwards and Futures 72

21 Yields and Forward Rates 496

5 Options 92

22 Interest Rate Swaps 538

6 Arbitrage and Trading 112

23 Single-Period Binomial Heath–Jarrow–Morton Model 558

7 Financial Engineering and Swaps 129

24 Multiperiod Binomial HJM Model 592

PART II Forwards and Futures 8 Forwards and Futures Markets 154 9 Futures Trading 172 10 Futures Regulations 192 11 The Cost-of-Carry Model 209

25 The Heath–Jarrow–Morton Libor Model 618 26 Risk Management Models 655 Appendix A: Mathematics and Statistics 684

12 The Extended Cost-of-Carry Model 229

Glossary 693

13 Futures Hedging 256

Notation 710

References 704

Additional Sources and Websites 712

PART III Options 14 Options Markets and Trading 288

Books on Derivatives and Risk Management 715

15 Option Trading Strategies 310

Name-Index 718

16 Option Relations 336

Subject-Index 720

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Contents

About the Authors VI Preface to Second Edition XXVII Preface to First Edition XXXI

PA R T I Introduction CHAPTER 1 Derivatives and Risk Management

.....................

2

1.1 Introduction 3 1.2 Financial Innovation 5 Expanding Derivatives Markets 5 Two Economic Motives 7 1.3 Traded Derivative Securities 7 EXTENSION 1.1: The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation 8 Diverse Views on Derivatives 9 Applications and Uses of Derivatives 10 A Quest for Better Models 12 1.4 Defining, Measuring, and Managing Risk 12 1.5 The Regulator’s Classification of Risk 12 1.6 Portfolio Risk Management 14 1.7 Corporate Financial Risk Management 14 Risks That Businesses Face 14 Nonhedged Risks 16 Risk Management in a Blue Chip Company 16 1.8 Risk Management Perspectives in This Book 17 1.9 Summary 18 1.10 Cases 19 1.11 Questions and Problems 19

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CHAPTER 2 Interest Rates

.............................................................. 22

2.1 Introduction 23 2.2 Rate of Return 24 2.3 Basic Interest Rates: Simple, Compound, and Continuously Compounded 25 EXTENSION 2.1: Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time 26 2.4 Discounting (PV) and Compounding (FV): Moving Money across Time 31 EXTENSION 2.2: Moving Multiple Cash Flows across Time 34 2.5 US Treasury Securities 37 2.6 US Federal Debt Auction Markets 38 2.7 Different Ways of Investing in Treasury Securities 40 The Treasury Auction and Its Associated Markets 40 The Repo and the Reverse Repo Market 41 EXTENSION 2.3: A Repurchase Agreement 41 Interest Rate Derivatives 43 2.8 Treasury Bills, Notes, Bonds, and STRIPS 43 2.9 Libor versus a Libor Rate Index 47 2.10 Summary 48 2.11 Cases 50 2.12 Questions and Problems 50

CHAPTER 3 Stocks

............................................................................. 53

3.1 Introduction 54 3.2 Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets 54 3.3 Brokers, Dealers, and Traders in Securities Markets 56 3.4 Automation of Trading 58 3.5 The Three-Step Process of Transacting Exchange-Traded Securities 58 3.6 Buying and Selling Stocks 59 Trading at the New York Stock Exchange 60 Over-the-Counter Trading 61 Alternative Trading Systems: Dark Pools and Electronic Communications Networks 61 3.7 Dollar Dividends and Dividend Yields 62 3.8 Short Selling Stocks 64

CONTENTS

3.9 Margin–-Security Deposits That Facilitate Trading 66 EXTENSION 3.1: Margin and Stock Trading 67 3.10 Summary 68 3.11 Cases 70 3.12 Questions and Problems 70

CHAPTER 4 Forwards and Futures

.............................................. 72

4.1 Introduction 73 4.2 Forward Contracts 73 4.3 The Over-the-Counter Market for Trading Forwards 78 4.4 Futures Contracts 81 4.5 Exchange Trading of a Futures Contract 83 EXTENSION 4.1: Futures Exchanges in China and India 86 4.6 Hedging with Forwards and Futures 88 4.7 Summary 89 4.8 Cases 90 4.9 Questions and Problems 90

CHAPTER 5 Options

.......................................................................... 92

5.1 Introduction 93 5.2 Options 93 5.3 Call Options 95 Call Payoffs and Profit Diagrams 95 The Call’s Intrinsic and Time Values 98 Price Bounds for American Calls 99 5.4 Put Options 100 Put Payoffs and Profit Diagrams 100 The Put’s Intrinsic and Time Values 102 Price Bounds for American Puts 103 5.5 Exchange-Traded Options 105 5.6 Longs and Shorts in Different Markets 107 5.7 Order Placement Strategies 107 5.8 Summary 108 5.9 Cases 109 5.10 Questions and Problems 109

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CHAPTER 6 Arbitrage and Trading

........................................... 112

6.1 Introduction 113 6.2 The Concept of Arbitrage 113 6.3 The No-Arbitrage Principle for Derivative Pricing 117 The Law of One Price 117 Nothing Comes from Nothing 117 6.4 Efficient Markets 118 6.5 In Pursuit of Arbitrage Opportunities 119 The Closed-End Fund Puzzle 119 Spread Trading 120 Index Arbitrage 121 EXTENSION 6.1: Stock Indexes 121 EXTENSION 6.2: Index Arbitrage 123 6.6 Illegal Arbitrage Opportunities 124 6.7 Summary 126 6.8 Cases 127 6.9 Questions and Problems 127

CHAPTER 7 Financial Engineering and Swaps

..................... 129

7.1 Introduction 130 7.2 The Build and Break Approach 130 7.3 Financial Engineering 131 Cash Flows versus Asset Values 131 Examples 132 7.4 An Introduction to Swaps 135 7.5 Applications and Uses of Swaps 135 7.6 Types of Swaps 136 Interest Rate Swaps 136 Forex Swaps 138 Currency Swaps 139 EXTENSION 7.1: Valuing Fixed-for-Fixed Currency Swaps 142 Commodity and Equity Swaps 146 Credit Default Swaps 147 7.7 Summary 148 7.8 Cases 149 7.9 Questions and Problems 149

CONTENTS

PA R T II Forwards and Futures CHAPTER 8 Forwards and Futures Markets

....................... 154

8.1 Introduction 155 8.2 Applications and Uses of Forwards and Futures 155 8.3 A Brief History of Forwards and Futures 156 Early Trading of Forward and Futures-Type Contracts 156 US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926 157 Recent Developments, 1970 Onward 159 8.4 Futures Contract Features and Price Quotes 161 Commodity and Financial Futures Contracts 161 The Gold Futures Contract 161 Gold Futures Price Quotes 164 The Exchange and Clearinghouse 166 8.5 Commodity Price Indexes 167 EXTENSION 8.1: Doomsters and Boomsters 168 8.6 Summary 169 8.7 Cases 170 8.8 Questions and Problems 170

CHAPTER 9 Futures Trading

...................................................... 172

9.1 Introduction 173 9.2 Brokers, Dealers, and the Futures Industry 173 9.3 Trading Futures 174 9.4 Margin Accounts and Daily Settlement 176 Daily Settlement 176 9.5 Futures and Forward Price Relations 180 Equality of Futures and Spot Price at Maturity 183 Equality of Forward and Futures Prices before Maturity 183 Convergence of the Basis 185 9.6 Trading Spreads 185 9.7 Summary 188 9.8 Cases 189 9.9 Questions and Problems 189

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CHAPTER 10 Futures Regulations

............................................. 192

10.1 Introduction 193 10.2 Which Markets Have Futures? 193 Price Discovery 194 Hedging 194 Speculation 195 EXTENSION 10.1: When Sellers Have Market Power 195 10.3 Regulation of US Futures Markets 196 Major Regulatory Acts 196 The CFTC, the NFA, and the Regulatory Role of Exchanges 199 Enforcement Actions by the CFTC and the NFA 201 10.4 Manipulation in Futures Markets 202 The Hunts Silver Case 202 Salomon Brothers and US Treasury Securities 203 Banging the Close 204 10.5 Managing Commodity Markets 205 10.6 Summary 206 10.7 Cases 206 10.8 Questions and Problems 207

CHAPTER 11 The Cost-of-Carry Model

.................................. 209

11.1 Introduction 210 11.2 A Cost-of-Carry Example 210 11.3 The Assumptions 214 11.4 The Cost-of-Carry Model 216 The Model Setup 216 Using the Law of One Price 217 Using Nothing Comes from Nothing 217 Different Methods for Computing Interest 218 The Arbitrage Table Approach 220 11.5 Valuing a Forward Contract at Intermediate Dates 221 11.6 Linking Forward Prices of Different Maturities 224 11.7 Summary 225 11.8 Cases 226 11.9 Questions and Problems 226

CONTENTS

CHAPTER 12 The Extended Cost-of-Carry Model

............. 229

12.1 Introduction 230 12.2 A Family of Forward Pricing Models 230 12.3 Forwards on Dividend-Paying Stocks 232 The Cost-of-Carry Model with Dollar Dividends 232 The Cost-of-Carry Model with a Dividend Yield 235 A Synthetic Index 236 The Foreign Currency Forward Price 239 12.4 Extended Cost-of-Carry Models 243 12.5 Backwardation, Contango, Normal Backwardation, and Normal Contango 246 12.6 Market Imperfections 249 12.7 Summary 252 12.8 Cases 253 12.9 Questions and Problems 253

CHAPTER 13 Futures Hedging

................................................... 256

13.1 Introduction 257 13.2 To Hedge or Not to Hedge 257 Should the Firm or the Individual Hedge? 257 The Costs and Benefits of Corporate Hedging 259 EXTENSION 13.1: Airlines and Fuel Price Risk 260 EXTENSION 13.2: A Hedged Firm Capturing a Tax Loss 262 13.3 Hedging with Futures 264 Perfect and Imperfect Hedges 264 Basis Risk 264 Guidelines for Futures Hedging 267 13.4 Risk-Minimization Hedging 268 The Mean-Variance Approach 268 Limitations of Risk-Minimization Hedging 272 13.5 Futures versus Forward Hedging 272 13.6 Spreadsheet Applications: Computing h 273 13.7 Summary 275 13.8 Appendix 276 Deriving the Minimum-Variance Hedge Ratio (h) 276 Computing the Minimum-Variance Hedge Ratio (h) 277

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Statistical Approach 277 Econometric Approach: A Linear Regression Model 281 13.9 Cases 281 13.10 Questions and Problems 282

PA R T III Options CHAPTER 14 Options Markets and Trading

......................... 288

14.1 Introduction 289 14.2 Exchange-Traded Options 289 14.3 A History of Options 290 Early Trading of Options 291 The Year 1973: The Watershed Year and After 293 EXTENSION 14.1: Options on Futures 295 14.4 Option Contract Features 296 Maturity Dates 297 Strike Prices 297 Margin Requirements and Position Limits 298 Dividends and Stock Splits 299 14.5 Options Trading, Exercising, and the Expiration Process 300 The Trading Process 300 Exercising the Option 300 The Expiration Process 301 14.6 Options Price Quotes 301 14.7 Regulation and Manipulation in Options Markets 305 14.8 Summary 306 14.9 Cases 307 14.10 Questions and Problems 307

CHAPTER 15 Option Trading Strategies 15.1 Introduction 311 15.2 Traders in Options Markets 311 15.3 Profit Diagrams 312 15.4 Options Strategies 313

................................ 310

CONTENTS

Common Options Price Data 313 Naked Trades: The Basic Building Blocks 314 15.5 Hedged Strategies 317 15.6 Spread Strategies 320 EXTENSION 15.1: Collars, Ratio Spreads, Butterflies, and Condors 323 EXTENSION 15.2: Reinsurance Contracts and Call Spreads 327 15.7 Combination Strategies 329 15.8 Summary 331 15.9 Cases 332 15.10 Questions and Problems 332

CHAPTER 16 Option Relations

................................................... 336

16.1 Introduction 337 16.2 A Graphical Approach to Put–Call Parity 337 16.3 Put–Call Parity for European Options 338 16.4 Market Imperfections 342 EXTENSION 16.1: Put–Call Parity in Imperfect Markets 343 EXTENSION 16.2: Dividends and American Options 344 16.5 Options Price Restrictions 346 EXTENSION 16.3: The Superglue Argument 352 16.6 Early Exercise of American Options 357 No Dividends 357 Dividends and Early Exercise 360 16.7 Summary 362 16.8 Cases 364 16.9 Questions and Problems 364

CHAPTER 17 Single-Period Binomial Model 17.1 Introduction 368 17.2 Applications and Uses of the Binomial Model 368 17.3 A Brief History of Options Pricing Models 369 Option Pricing Pre-1973 369 Options Pricing, 1973 and After 370 17.4 An Example 372

........................ 367

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17.5 The Assumptions 374 17.6 The Single-Period Model 376 The No-Arbitrage Principle 376 Building Binomial Trees 377 Arbitrage-Free Trees 378 Stock Prices and Martingales 378 The Pricing Model 379 The Hedge Ratio 380 Risk-Neutral Valuation 381 Actual versus Pseudo-probabilities 382 17.7 Summary 383 17.8 Appendix 384 Proving the No-Arbitrage Argument 384 The Probabilities and Risk Premium 385 17.9 Cases 388 17.10 Questions and Problems 388

CHAPTER 18 Multiperiod Binomial Model

............................ 392

18.1 Introduction 393 18.2 Toward a Multiperiod Binomial Option Pricing Model 393 The Stock Price Evolution 393 Binomial Option Price Data 394 The Stock Price Tree 395 18.3 A Two-Period Binomial Model 396 Backward Induction 396 Option Pricing via Synthetic Construction (Method 1) 397 Repeat, Repeat: Risk-Neutral Pricing (Method 2) 400 One-Step Valuation: Prelude to the Multiperiod Model (Method 3) 401 18.4 The Multiperiod Binomial Option Pricing Model 403 Binomial Coefficients and Pseudo-probabilities 403 Recasting the Two-Period Example in the Multiperiod Framework 403 The n-Period Binomial Option Pricing Model 405 EXTENSION 18.1: Linking the Binomial Model to the Black–Scholes–Merton Model 407 18.5 Extending the Binomial Model 409 Known Dividends 409 Valuing American Options 410

CONTENTS

18.6 Spreadsheet Applications 413 A Two-Period Binomial Example 413 A Sixteen-Period Example 416 18.7 Summary 418 18.8 Cases 419 18.9 Questions and Problems 419

CHAPTER 19 The Black–Scholes–Merton Model

............... 422

19.1 Introduction 423 19.2 Nobel Prize–Winning Works (1973) 423 19.3 The Assumptions 424 EXTENSION 19.1: Bubbles and Option Pricing 424 19.4 The Pricing and Hedging Argument 428 19.5 The Black–Scholes–Merton Formula 429 19.6 Understanding the Black–Scholes–Merton Model 432 Stock Prices and Martingales 432 Risk-Neutral Valuation 433 Actual versus Pseudo-probabilities 433 19.7 The Greeks 434 Interpreting the Greeks 434 Some Road Bumps Ahead 439 EXTENSION 19.2: Market Manipulation and Option Pricing 439 19.8 The Inputs 440 Observable Inputs 441 Volatility: The Elusive Input 441 19.9 Extending the Black–Scholes–Merton Model 444 Adjusting for Dividends 444 Foreign Currency Options 446 Valuing American Options 447 EXTENSION 19.3: Exotic Options 448 19.10 Summary 449 19.11 Appendix 451 Modeling the Stock Price Evolution 451 Continuously Compounded (Logarithmic) Return 451 Introducing Uncertainty 452 The Central Limit Theorem 453

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First Derivation of the Black–Scholes–Merton Formula (as a Limit of the Binomial Model) 454 Second Derivation of the Black–Scholes–Merton Formula (Using Risk-Neutral Valuation) 457 The Probabilities and the Risk Premium 457 Proof of (24) 458 19.12 Cases 459 19.13 Questions and Problems 459

CHAPTER 20 Using the Black–Scholes–Merton Model 20.1 Introduction 463 20.2 Hedging the Greeks 463 Delta Hedging 463 Gamma Hedging 466 20.3 Hedging a portfolio of options 469 20.4 Vega Hedging 470 EXTENSION 20.1: Stochastic Volatility Option Models 473 20.5 Calibration 474 Theoretical and Econometric Models 474 Using Calibration 475 Implied Volatilities 477 EXTENSION 20.2: VIX: The Fear Index 480 20.6 The Black–Scholes–Merton Model: A Postscript 480 20.7 Summary 481 20.8 Appendix 482 The Mathematics of Delta, Gamma, and Vega Hedging 482 A Third Derivation of the Black–Scholes–Merton Formula 486 20.9 Cases 487 20.10 Questions and Problems 488 Overview 490 Data (from the Wall Street Journal, March 3, 2000) 490 The Project 491

... 462

CONTENTS

PA R T IV Interest Rate Derivatives CHAPTER 21 Yields and Forward Rates

.................................. 496

21.1 Introduction 497 EXTENSION 21.1: Home Ownership and Derivatives 498 21.2 Yields 499 Revisiting Zeros 499 Yields and the Yield Curve 500 EXTENSION 21.2: The Expectations Hypothesis 504 21.3 The Traditional Approach 507 Duration 507 Modified Duration Hedging 510 Applications and Limitations 512 21.4 Forward Rates 513 The Definition 514 Understanding Forward Rates 516 Using Forward Rates 517 EXTENSION 21.3: Computing Forward Rates from Coupon Bond Prices 519 21.5 The Basic Interest Rate Derivatives Contracts 520 A Brief History 520 Forward Rate Agreements 521 Interest Rate Futures 524 EXTENSION 21.4: Treasury Futures 529 The Equivalence between Forward and FRA Rates 530 21.6 Summary 531 21.7 Cases 534 21.8 Questions and Problems 535

CHAPTER 22 Interest Rate Swaps

............................................. 538

22.1 Introduction 539 22.2 A Brief History 539 The Introduction of Swap Contracts 540 From a Brokerage to a Dealership Market 541 ISDA and Standardization of Contracts 541

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CONTENTS

22.3 Institutional Features 542 Entering a Swap Contract 542 Documentation 543 Closing a Swap Position 544 22.4 Valuation 544 22.5 Variations of Interest Rate Swaps 549 22.6 Swaps and FRAs 550 Synthesizing Swaps with Eurodollars and FRAs 550 The Yield and Swap Curve 552 EXTENSION 22.1: Computing Forward Rates from Swap Rates 553 22.7 Summary 555 22.8 Cases 555 22.9 Questions and Problems 555

CHAPTER 23 Single-Period Binomial Heath–Jarrow–Morton Model

......................... 558

23.1 Introduction 559 23.2 The Basic Interest Rate Derivatives 560 Uses of Caps, Floors, and Collars 561 23.3 A Brief History of Interest Rate Derivatives Models 564 23.4 The Assumptions 565 23.5 The Single-Period Model 570 Arbitrage-Free Evolutions 570 Zero-Coupon Bond Prices and Martingales 573 Understanding the Equal Pseudo-probability Condition 575 Actual versus Pseudo-probabilities 576 Caplet Pricing 576 The Hedge Ratio (the Holy Grail) 581 Risk-Neutral Valuation 581 Valuing a Floorlet 582 Valuing Various Interest Rate Derivatives 584 Multiple Factors 584 23.6 Summary 585 23.7 Appendix 586 The Equal Pseudo-probability Condition 586 Proof of Sufficiency 587 Proof of Caplet and Floorlet Parity 588

CONTENTS

23.8 Cases 589 23.9 Questions and Problems 589

CHAPTER 24 Multiperiod Binomial HJM Model

................. 592

24.1 Introduction 593 24.2 The Assumptions 593 24.3 The Two-Period Model 597 Arbitrage-Free Evolutions 597 Zero-Coupon Bond Prices and Martingales 599 Caplet Pricing 601 Floorlet Pricing 605 Valuing Various Interest Rate Derivatives 606 Multiple Factors 606 24.4 The Multiperiod Model 607 24.5 Forwards, Futures, and Swaptions 607 Forward Rate Agreements 607 Eurodollar Futures 608 Comparing Forward and Futures Rates 610 Swaptions 611 24.6 Summary 612 24.7 Appendix 613 Linking the Binomial HJM Model and the HJM Libor Model 613 24.8 Cases 615 24.9 Questions and Problems 616

CHAPTER 25 The Heath–Jarrow–Morton Libor Model

.... 618

25.1 Introduction 619 25.2 Why Caplets? 619 25.3 A History of Caplet Pricing 520 Black’s Model 620 The Heath–Jarrow–Morton Model 621 The HJM Libor Model 621 EXTENSION 25.1: Black’s Model for Pricing Commodity Options and Caplets 621 25.4 The Assumptions 622

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25.5 Pricing Caplets 627 Caplet and Floorlet Pricing (CFP) Data 627 Caplet Pricing Model 627 25.6 The Inputs 631 Observable Inputs 631 The Average Forward Rate Volatility 632 25.7 Understanding the HJM Libor Model 637 Forward Rates and Martingales 637 Risk-Neutral Valuation 637 Actual versus Pseudo-probabilities 638 25.8 Caps and Floors 638 Caps 639 Floorlets 639 Caplet–Floorlet Parity 639 Floors 641 25.9 Using the HJM Libor Model 641 The Greeks 642 Delta Hedging 643 Gamma Hedging 646 25.10 American Options, Futures, Swaptions, and Other Derivatives 647 25.11 Summary 647 25.12 Appendix 649 25.13 Cases 652 25.14 Questions and Problems 652

CHAPTER 26 Risk Management Models

................................ 655

26.1 Introduction 656 26.2 A Framework for Financial Risk Management 658 26.3 Computing the Loss Distribution 660 26.4 Value-at-Risk and Scenario Analysis 662 Value-at-Risk 662 Scenario Analysis 666 EXTENSION 26.1: Risk Measures 667 26.5 The Four Risks 668 Market Risk 668 Credit Risk 669 Structural Models 670

CONTENTS

EXTENSION 26.2: Real Options 672 Reduced-Form Models 673 EXTENSION 26.3: Credit Default Swaps 676 Liquidity Risk 677 Operational Risk 678 26.6 The Future of Models and Traded Derivatives 679 Model Risk 679 Derivatives 680 26.7 Summary 680 26.8 Cases 681 26.9 Questions and Problems 681

APPE N DI X A Mathematics and Statistics A.1 Exponents and Logarithms 685 Exponents 685 Logarithms 685 Continuous Compounding 687 A.2 The Binomial Theorem 687 A.3 The Normal Distribution 688 A.4 A Taylor Series Expansion 691 Single Variables 691 Multiple Variables 692 Glossary 693 References 704 Notation 710 Additional Sources and Websites 712 Books on Derivatives and Risk Management 715 Name-Index 718 Subject-Index 720

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Preface to Second Edition

We are pleased by the global reception of the first edition. The book has seen classroom usage in over 100 courses in 75 universities in 22 countries. It has been used widely as: (a) an introductory textbook for undergraduates (in business schools, in Arts & Sciences, as well as other programs), (b) an introductory text for MBAs, MS, and other masters students, (c) a recommended text in derivatives courses, and (d) a reference text in specialized courses.

Changes in the Second Edition Based on feedback from colleagues, students, and our own teaching experience we focused on updating institutional facts, reducing the historical and institutional material, and cleaning typos. A Solutions Manual for end-of-chapter Questions and Problems remains available for lecturers, together with PowerPoint slides and a Test Bank. These ancillaries can be downloaded from https://www.worldscientific.com/worldscibooks/ 10.1142/y0018-sm Although our free-software PRICED! still works in most computers, we find that many professors and students are more comfortable developing their own pricing models through Excel (we also make our Excel files available to adopters). Consequently, we continue to make PRICED! available but we do not provide any support. Lecturers and students who wish to download PRICED! can visit https://www.worldscientific.com/worldscibooks/10.1142/y0018 and click on the “Supplementary” tab.

Acknowledgements We owe many thanks to colleagues and students for their help, support, and suggestions. We especially thank two colleagues for their extraordinary help and support: Scott Fung (California State University, East Bay) and Thijs van der Heijden (University of Melbourne). We also thank Binay Bhushan Chakrabarti, Prem Chandrani, Ronald Ehrenberg, Craig Holden, Sreenivas Kamma, Ravindra Patil, and Zhenyu Wang for their comments, help, and support. We thank Yu Yan who translated our book into Chinese.

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PREFACE TO SECOND EDITION

We thank the following colleagues who, our records show, have adopted the book for classroom use as a required or recommended reading: Sofiane Aboura, Université de Paris XIII, Sorbonne Paris Cité; Vedat Akgiray, Bogaziçi ̆ Üniversitesi; Ali Akyol, University of Melbourne; Joao Freire de Andrade, Universidade NOVA de Lisboa; Kostas Andriosopoulos, ESCP Europe, London; Kevin Aretz, University of Manchester; C Bülent Aybar, Southern New Hampshire University; Giovanni Barone-Adesi, Università della Svizzera italiana at Lugano; Basabi Bhattacharya, Jadavpur University; Nalinaksha Bhattacharyya, University of Alaska Anchorage; Jedrzej Bialkowski, University of Canterbury; Ben Chongyang Chen, University of Memphis; Hsin-Fu Chen, Lunghwa University of Science & Technology; William Cheung, Waseda University; Paul Moon Sub Choi, Ewha Womans University; Steven P Clark, University of North Carolina at Charlotte; Francisco José Climent Diranzo, Universitat de València; Vladislav Damjanovic, University of Essex; Theodoros Diasakos, University of Stirling; Sina Erdal, University of Essex; Shih-Chuan Fang, Chaoyang University of Technology; Scott Fung, California State University East Bay; Gianluca Fusai, City, University London; Neal Galpin, University of Melbourne; George P Gao, T Rowe Price; Rajesh Ghai, University of Texas at Dallas; Chinmoy Ghosh, University of Connecticut; Bruce Grundy, University of Melbourne; Anthony Hall, University of Technology, Sydney; Joongho Han, Sungkyunkwan University; Barry Harrison, Nottingham Trent University; Mia Hinnerich, Stockholms Universitet; Darien Huang, Cornell University; Nagaratnam Jeyasreedharan, University of Tasmania; Timothy C Johnson, Heriot-Watt University; Jeffrey Jolley, California State University Fullerton; Andreas Kaeck, University of Sussex; Hui-Chuan Kao, Ling Tung University; Cenk Cevat Karahan, Bogaziçi ̆ Üniversitesi; Roger Klee, Cleveland State University; Ioannis Kyriakou, City, University London; Pascal Letourneau, University of Wisconsin-Whitewater; Hung-Chi Li, National Cheng Kung University;

PREFACE TO SECOND EDITION

Hai Lin, Victoria University of Wellington; Ravi Mateti, Concordia University; João Amaro de Matos, Universidade NOVA de Lisboa; Hongghi Min, Korea Advanced Institute of Science and Technology; Nikos Nomikos, City, University London; Kurtay Ogunc, Texas A & M University-Commerce; Jacques Olivier, HEC Paris; Late Philip Palm, who was at University of Washington Bothell; Ging-Ginq Pan, National Pingtung University of Science and Technology; Christophe Perignon, HEC Paris; Ser-Huang Poon, University of Manchester; Marcel Prokopczuk, Zeppelin University; Paulo J M Rodrigues, Universiteit Maastricht; Juan Carlos Rodriguez, Tilburg University; Mehmet Saglam, ̆ University of Cincinnati; Rafael de Santiago, Universidad de Navarra; Ravi Sastry, Southern Methodist University; Norman Seeger, Vrije Universiteit Amsterdam; Elena Sernova, HEC Lausanne; Kaisheng Song, University of Texas at Dallas; Erling Steigum, BI Norwegian Business School; Liya Shen, University of Essex; Manouchehr Tavakoli, University of St Andrews; Siegfried Trautmann, Johannes Gutenberg Universitat Mainz; Jiao Tong, Robert Gordon University; Hui-Huang Tsai, National United University; Jen-Tsung Tsu, Chien Hsin University of Science and Technology; Calum Turvey, Cornell University; Eleuterio Vallelado Gonzalez, Universidad de Valladolid; Thijs van der Heijden, University of Melbourne; Dmitry Vedenov, Texas A & M University; Lee Wakeman, Ohio University; Nathan Walcott, Southern Methodist University; Costas Xiouros, BI Norwegian Business School; Lulu Zeng, Purdue University; J Feng Zhao, University of Texas at Dallas; Lin Zou, Texas Woman’s University. We thank our editor Jack Repcheck of W. W. Norton & Company and his colleagues Dorothy Cook, Edward Crutchley, Theresia Kowara, Lindsey Osteen, Lauren Quantrill, Eric Svendsen, Janise Turso, Dina Vakser, and others who have faciliated the global acceptance that the book enjoys today. After Jack’s sudden and sad death, we moved to World Scientific Publishing. Our special thanks to our new editor Yubing Zhai and her colleagues Shreya Gopi, Karimah Samsudin and others who have made possible a smooth transition and a fine production of the second edition.

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Preface to First Edition

“History has many cunning passages, contrived corridors,” wrote T. S. Eliot in the poem “Gerontion.” The history of options and futures can be so described. Derivatives have traded for centuries in small, over-the-counter markets in London, New York, and several cities of continental Europe. In 1688, a sophisticated rice futures market was established in Osaka, Japan, that thrived for 250 years. Modern commodity futures markets began in 1848 with the founding of the Chicago Board of Trade to help protect farmers from commodity price swings. After many placid decades, the calm was again broken with the inauguration of the world’s first financial futures market in 1972 at the Chicago Mercantile Exchange and in 1973 with the opening of the Chicago Board Options Exchange. Serendipitously, 1973 also saw the publication of the Nobel Prize–winning Black–Scholes–Merton (BSM) option pricing model, which further spurred market expansion by enabling better pricing and hedging. Sometimes viewed as harmful and sometimes viewed as beneficial by the financial press and the public, derivatives have nonetheless played a significant role in financial markets across the centuries. Following the 2007 credit crisis, derivatives are most recently playing the role of the harmful security again, and new financial regulations have been proposed to rein them in. Although perhaps well intentioned, these attacks on derivatives are hurled by those unfamiliar with their proper use. Derivative securities, if used properly, reduce risk and facilitate real economic growth in the economy. In today’s complex world, modern financial institutions cannot succeed without the use of derivatives for managing the varied risks of their assets and liabilities. An understanding of their proper usage comes through a careful study of derivatives, which brings us to the purpose for writing this book. This book was written to be the first book read on derivatives and not the last. Our aim has been to design a book that is closely connected to real markets, examines the uses of derivatives but warns against their abuses, and presents only the necessary quantitative material in an easily digestible form—and no more! Given our purpose, this book differs from all existing derivatives textbooks in several important ways: ■

First, it is an introduction. We wanted to create a textbook accessible to MBAs and undergraduates both in terms of the concepts and mathematics. Option pricing is normally thought of as a complex, mathematical, and difficult topic. Our experience is that this topic can be presented simply and intuitively.

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Second, it is about financial markets. We wanted to write an economics, not a quantitative, book on derivatives. Since the first textbooks on option pricing,1 derivatives textbooks have taken a quantitative approach to the topic and are often encyclopedic in presentation. Little if any effort was spent on the underlying economics. In contrast, an economics perspective relates the market structure to the assumptions underlying the models. An understanding of when to use and when not to use a model based on its assumptions must be included. Our book does this.

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Third, it is a book about risk management. Generally speaking, there are four risks to be managed: (1) market risk, which includes commodity (including equity) price risk, interest rate risk, and foreign currency price risk; (2) credit risk; (3) liquidity risk; and (4) operational risk. We walk you through these different risks, with an emphasis on market risk, and discuss how they can be managed in business as well as in one’s personal life.

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Fourth, not only do governments regulate the markets, but many government entities use derivatives to promote the public’s welfare. The book often discusses the relevant issues from a regulator’s point of view and from a public policy perspective.

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Fifth, another unique feature of this book is an intuitive and accessible presentation of the Heath–Jarrow–Morton (HJM) model, which is the most advanced as well as a widely used derivatives pricing model. To make this model accessible, we first present the classical option pricing theory centering on the BSM model in a userfriendly fashion. Our presentation of the BSM model, however, is done with an eye toward the HJM model, emphasizing those aspects of the BSM model that are needed later. Then, when we study the HJM model, less development is needed. This approach enables us to present the HJM model in a parallel fashion to that of the BSM model, so if you see one, you see them both!

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Sixth, option pricing has a fascinating history, filled with colorful people and events. We share this history with the reader. This history is obtained from century-old books (now, uniquely available via the Internet), recent books, newspaper and magazine articles, websites, and personal experiences.

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Seventh, we have included enrichment material for the advanced reader through the use of inserts and appendixes. Many of these inserts include current research insights not available in existing textbooks.

The organization of this book is purposely designed to facilitate its use in many courses relating to options, futures, derivatives, risk management, investments, fixed income securities, and financial institutions. Three major courses that can be taught from this book are: (1) derivatives, (2) futures and commodities, and (3) fixed income securities and interest rate derivatives. In a sense, there are many books within this one cover.

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Jarrow and Rudd (1983) and Cox and Rubinstein (1984).

PREFACE TO FIRST EDITION

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Derivatives. This is a standard course on basic derivatives that gives an introduction to forwards, futures, options, and swaps but excludes interest rate derivatives for the most part. For this, use chapters 1–20 and most of Chapter 22. This would be for first-year MBAs and masters of financial engineering and for upper-level undergraduates in business schools, engineering schools, and economics departments.

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Futures and commodities. This is a course on futures and commodities, excluding option pricing. For this, use Chapters 1–6 and 8–13. The target audience is as for the prior item.

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Fixed-income securities and interest rate derivatives. This would be a models-based approach to teaching this material. Integrating ongoing research developments, Jarrow has been teaching such a course at Cornell University for over twenty years. For this, use Chapters 1, 2, 4, 6, 9, and 15–26. This course can be taught to MBAs and more mathematically inclined upper level undergraduates.

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Case-based courses. We have also recommended cases at the end of each chapter so that instructors can easily develop a case study–oriented derivatives course.

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Spreadsheet-based courses. Unlike other textbooks, we do not provide black boxes in which you input data and get a derivative price! In line with current teaching trends, we have woven spreadsheet applications throughout the text. Our aim is for students to achieve self-sufficiency so that they can generate all the models and graphs in this book via Excel. In addition, spreadsheet software—called Priced!—especially designed for this textbook is available to facilitate learning and to teach the course material.

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Use in courses in other areas. Courses in accounting and law can use this as background material.

This book is a “download” of our understandings of markets and derivatives obtained from decades of research, teaching, and consulting. The material has been class tested at the Cornell University, Helsinki School of Economics and Business Administration, Hong Kong University of Science and Technology, the Indian Institutes of Management at Ahmedabad and at Calcutta, Indiana University (Bloomington), University of Colorado at Boulder, and the University of North Carolina at Chapel Hill. We hope that the reader will have a better understanding of derivative securities and risk management models after reading this book. Although we have tried to make this textbook error free, please notify us if you discover any errors.

The Ancillaries This text is accompanied by a number of important ancillaries, each intended to enhance the learning experience for the student and the teaching experience for the instructor.

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For Students and Instructors PRICED! Developed by a Cornell computer science Ph.D. Tibor Jánosi specifically for this text and in collaboration with book authors Robert Jarrow and Arkadev Chatterjea, Priced! is Excel-based software that makes the models real rather than hypothetical. The software computes prices and hedge ratios for the four key derivative security models contained in the book: (1) the binomial model, (2) the Black–Scholes– Merton (BSM) model, (3) the discrete time Heath–Jarrow–Morton (HJM) model, and (4) the HJM Libor model. The prices and hedge ratios are represented visually—both graphically and in trees—for easy analysis by students and instructors. For quick recognition and clarity, the software’s inputs and outputs are color-coded. Priced! is also completely dynamic; when changes are made to inputs, outputs are updated instantaneously. This software can be used to illustrate all of the key concepts associated with the derivative models discussed in the course. Instructors can illustrate how a model’s prices and hedge ratios change when inputs are changed. Students can even use Priced! and current market prices from the financial press to compute actual prices.

For Students SOLUTIONS MANUAL Written entirely by the text’s authors, Robert Jarrow and Arkadev Chatterjea, the solutions manual provides completely worked solutions for all the problems included with the book. ISBN: 978-0-393-92094-9

For Instructors POWERPOINTS Created by coauthor Arkadev Chatterjea. The slides include lecture slides and all art from the book. There is also a separate set of PowerPoints created by Robert Jarrow for a fixed income course based on selected chapters in the book. Downloadable from wwnorton.com/instructors. TEST BANK Written by the text’s authors, Robert Jarrow and Arkadev Chatterjea. Downloadable formats available on wwnorton.com/instructors. ■

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Acknowledgments Bob thanks Peter Carr, Dilip Madan, Philip Protter, Siegfried Trautmann, Stuart Turnbull, and Don van Deventer for many conversations about derivatives over the years. For the many helpful discussions, comments, and material that helped improve this book, Bob and Arka would both like to thank Jeffery Abarbanell, Sobhesh K. Agarwalla, Jiyoun An, Warren B. Bailey, Gregory Besharov, Sommarat Chantarat,

PREFACE TO FIRST EDITION

Surjamukhi Chatterjea, Soikot Chatterjee, Sudheer Chava, Paul Moon Sub Choi, Steve Choi, Hazem Daouk, Werner Freystätter, Suman Ganguli, Nilanjan Ghosh, Michael A. Goldstein, Jason Harlow, Philip Ho, Michael F. Imhoff, Keon Hee Kim, Robert C. Klemkosky, Junghan Koo, Hao Li, Banikanta Mishra, Debi P. Mohapatra, Gillian Mulley, B. V. Phani, George Robinson, Ambar Sengupta, Asha Ram Sihag, Yusuke Tateno, and Han Zheng. We thank our family and friends for supporting us during this long project. In particular, Kaushik Basu, Nathaniel S. Behura, Amitava Bose, Alok Chakrabarti, Jennifer Conrad, Judson Devall, Ram Sewak Dubey, David Easley, Diego Garcia, Robert H. Jennings, Surendra Mansinghka, Robert T. Masson, Indranil Maulik, Tapan Mitra, Peter D. McClelland, Uri M. Possen, Erik Thorbecke, and Daniel J. Wszolek. We were fortunate to have a number of highly conscientious formal reviewers of the manuscript as it was taking shape. Their astute advice had a major impact on the final version of the manuscript. We cannot thank these reviewers enough: Farid AitSahlia (University of Florida), Gregory W. Brown (The University of North Carolina at Chapel Hill), Michael Ferguson (University of Cincinnati), Stephen Figlewski (New York University), Scott Fung (California State University, East Bay), Richard Rendleman (Dartmouth College), Nejat Seyhun (University of Michigan), Joel Vanden (The Pennsylvania State University), Kelly Welch (The University of Kansas), and Youchang Wu (University of Wisconsin– Madison). Finally, but certainly not least, special thanks to the editorial staff at Norton who have helped turn our manuscript into the finished book you are now reading: Hannah Bachman, Cassie del Pilar, Jack Repcheck, Nicole Sawa, and Amy Weintraub.

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I

Introduction

CHAPTER 1

Derivatives and Risk Management

CHAPTER 2

Interest Rates

CHAPTER 3

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Stocks

CHAPTER 6

Arbitrage and Trading

CHAPTER 7

Financial Engineering and Swaps

CHAPTER 4

Forwards and Futures

CHAPTER 5

Options

1 Derivatives and Risk Management 1.1 Introduction Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

1.2 Financial Innovation

1.5 The Regulator’s Classification of Risk

Expanding Derivatives Markets

1.6 Portfolio Risk Management

Two Economic Motives

1.7 Corporate Financial Risk Management

1.3 Traded Derivative Securities EXTENSION 1.1 The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation Diverse Views on Derivatives Applications and Uses of Derivatives A Quest for Better Models

1.4 Defining, Measuring, and Managing Risk

Risks That Businesses Face Nonhedged Risks Risk Management in a Blue Chip Company

1.8 Risk Management Perspectives in This Book 1.9 Summary 1.10 Cases 1.11 Questions and Problems

INTRODUCTION

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1.1

Introduction

The bursting of the housing price bubble, the credit crisis of 2007, the resulting losses of hundreds of billions of dollars on credit derivatives, and the failure of prominent financial institutions have forever changed the way the world views derivatives. Today derivatives are of interest not only to Wall Street but also to Main Street. Derivatives are cursed as one of the causes of the Great Recession of 2007–2009, a period of decreased economic output and high unemployment. But what are derivatives? A derivative security or a derivative is a financial contract that derives its value from an underlying asset’s price, such as a stock or a commodity, or even from an underlying financial index like an interest rate. A derivative can both reduce risk, by providing insurance (which, in financial parlance, is referred to as hedging), and magnify risk, by speculating on future events. Derivatives provide unique and different ways of investing and managing wealth that ordinary securities do not. Derivatives have a long and checkered past. In the 1960s, only a handful of individuals studied derivatives. No academic books covered the topic, and no college or university courses were available. Derivatives markets were small, located mostly in the US and Western Europe. Derivative users included only a limited number of traders in futures markets and on Wall Street. The options market existed as trading between professional traders (called the over-the-counter [OTC] market) with little activity. In addition, cheating charges often gave the options market disrepute. Derivatives discussion did not add sparkle to cocktail conversations, nor did it generate the allegations and condemnations that it does today. Brash young derivatives traders who drive exotic cars and move millions of dollars with the touch of a computer key didn’t exist. Although Einstein had developed the theory of relativity and astronauts had landed on the moon, no one knew how to price an option. That’s because in the 1960s, nobody cared, and derivatives were unimportant. What a difference the following decades have made! Beginning in the early 1970s, derivatives have undergone explosive growth in the types of contracts traded and in their importance to the financial and real economy. According to the Bank for International Settlements (known as the BIS), the markets are now global and measured in trillions of dollars. Indeed, as depicted in Figure 1.1, in December 2016 the total outstanding US dollar notional value for exchange-traded derivatives was (26,172 + 41,072 =) 67,245 billion and for OTC derivatives a staggering 500,419 billion. Hundreds of academics study derivatives, and thousands of articles have been written on the topic of pricing derivatives. Colleges and universities now offer numerous derivatives courses using textbooks written on the subject. Derivatives experts are in great demand. In fact, Wall Street firms hire PhDs in mathematics, engineering, and the natural sciences to understand derivatives—these folks are admirably called “rocket scientists” (“quants” is another name). If you understand derivatives, then you know cool stuff; you are hot and possibly dangerous. Today understanding derivatives is an integral part of the knowledge needed in the risk management of financial institutions.

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CHAPTER 1: DERIVATIVES AND RISK MANAGEMENT

FIGURE 1.1: Global Derivatives Market Derivatives Notional Amounts (December 2016, in billions of US dollars) 10015 1350 6140

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27864

26172 41073 68598

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386356

Futures (Exchange-Traded)

Options (Exchange-Traded)

Foreign Exchange Contracts (Global OTC)

Interest Rate Contracts (Global OTC)

Equity-Linked Contracts (Global OTC)

Commodity Contracts (Global OTC)

Credit Derivatives (Global OTC)

Other Derivatives (Global OTC)

Unallocated (Global OTC) Source: https://www.bis.org/statistics/extderiv.htm, Table D1: Exchange-traded futures and options, by location of exchange; and https://www.bis.org/statistics/derstats.htm, Table D5: Global OTC derivatives market.

Markets have changed to accommodate derivatives trading in three related ways: the introduction of new contracts and new exchanges, the consolidation and linking of exchanges, and the introduction of computer technology. Sometimes these changes happened with astounding quickness. For example, when twelve European nations replaced their currencies with the euro in 2002, financial markets for eurodenominated interest rate derivatives sprang up almost overnight, and in some cases, they quickly overtook the dollar-denominated market for similar interest rate derivatives. This chapter tells the fascinating story of this expansion in derivatives trading and the controversy surrounding its growth. An understanding of the meaning of financial risk is essential in fully understanding this story. Hence a discussion of financial risk comes next, from the regulator’s, the portfolio manager’s, and the corporate financial manager’s points of view. We explain each of these unique perspectives,

FINANCIAL INNOVATION

using them throughout the book to increase our understanding of the uses and abuses of derivatives. A summary completes the chapter.

1.2

Financial Innovation

Derivatives are at the core of financial innovation, for better or for worse. They are the innovations to which columnist David Wessel’s Wall Street Journal article titled “A Source of Our Bubble Trouble,” dated January 17, 2008, alludes:

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Modern finance is, truly, as powerful and innovative as modern science. More people own homes—many of them still making their mortgage payments—because mortgages were turned into securities sold around the globe. More workers enjoy stable jobs because finance shields their employers from the ups and downs of commodity prices. More genius inventors see dreams realized because of venture capital. More consumers get better, cheaper insurance or fatter retirement checks because of Wall Street wizardry.

Expressed at a time when most of the world was in the Great Recession, this view is challenged by those who blame derivatives for the crisis. Indeed, this article goes on to say that “tens of billions of dollars of losses in new-fangled investments [in derivatives and other complex securities] at the largest US financial institutions—and the belated realization that some of those Ph.D. wielding, computer-enhanced geniuses were overconfident in the extreme—strongly suggests some of the brainpower drawn to Wall Street would have been more productively employed elsewhere in the economy.” But derivatives have been trading in various guises for over two thousand years. They have continued to trade because, when used properly, they enable market participants to reduce risk from their portfolios and to earn financial rewards from trading on special skills and information. Indeed, derivatives help to advance or postpone cash flows (borrowing and lending), to accumulate wealth (saving), to protect against unfavorable outcomes (insurance or hedging), to commit funds to earn a financial return (investment), and to accept high risks in the hope of big returns (speculation or gambling), which often goes along with magnifying the scale of one’s financial returns (leverage). Financial markets grow and real economic activity prospers because derivatives make financial markets more efficient. This is a theme to which we return repeatedly throughout the book.

Expanding Derivatives Markets Many factors have fueled the growth of derivatives markets. These include regulatory reforms, an increase in international commerce, population growth, political changes, the integration of the world’s economy, and revolutionary strides in information technology (IT). The interrelated financial markets are now more susceptible to global shocks and financial crises. The financial world has become a mad, bad, and dangerous place—financially speaking! More pronounced business cycles, default by sovereign nations, high-risk leveraged bets by hedge funds, imprudent investment in complicated securities by unsophisticated investors, and fraudulent actions by rogue traders have the potential to shake financial markets to their core. Financial

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6

CHAPTER 1: DERIVATIVES AND RISK MANAGEMENT

regulators exist to help prevent these catastrophes from happening. And if used properly, derivatives can also help to mitigate their effects on aggregate wealth. To help achieve economic stability, the central bank of the US, the Federal Reserve System (often referred to as the Federal Reserve or simply the Fed), historically used monetary policy tools to keep interest rates stable. In 1979, the Fed also began targeting money supply growth. Despite this oversight, oil shocks and other supply-side disturbances created double-digit inflation rates in the 1970s and 1980s, which in turn led to double-digit US interest rates that wiggled more than ever before. These highly volatile interest rates created a need for securities to help corporations hedge this risk. Interest rate derivatives arose. The Chicago Board of Trade (CBOT; now part of the CME Group) developed the first interest rate derivative contract, the Ginnie Mae futures, in 1975 and the highly popular Treasury bond futures in 1977. The foreign exchange market is one of the world’s largest financial markets, where billions of dollars change hands daily. From the mid-1940s until the early 1970s, the world economy operated under the Bretton Woods system of fixed exchange rates—all the currencies were pegged to the US dollar, and the dollar was pegged to gold at $35 per ounce. This stable monetary system worked well for decades. However, problems arose when gold prices soared. Countries converted their currencies into dollars and bought cheap gold from the US at the bargain price of $35 per ounce, making huge profits. This was an arbitrage, a trade that makes riskless profits with no investment. Consequently, US gold reserves suffered a terrible decline. Because all currencies were tied to the dollar, the United States could not adjust the dollar’s exchange rate to fix the problem. Instead, US president Richard Nixon abandoned the Bretton Woods system in 1971. Currencies now float vis-à-vis one another in a so-called free market, although their values are frequently managed by central banks. Floating exchange rates are more volatile than fixed exchange rates, and to hedge this newly created currency risk, the huge foreign exchange derivatives market was created. In this regard, 1972 saw the opening of the International Monetary Market, a division of the Chicago Mercantile Exchange (CME or Merc; now part of the CME Group) to trade foreign currency futures. The world’s first exchange-traded financial derivatives contract was born! Given these regulatory changes and well-functioning interest rate and foreign currency derivatives markets, in the recent past, many economists believed that a new era of greater macroeconomic stability had dawned, dubbed the Great Moderation. During the two decades before the new millennium, fluctuations in the growth of real output and inflation had declined, stock market volatility was reduced, and business cycles were tamed. However, the tide soon turned. In 2007–2009, many nations were mired in the Great Recession, with declining economic output and large unemployment. Stock market volatility, as measured by the widely followed VIX Index, shot up from 10 percent to an astonishing 89.53 percent in October 2008.1 Volatility had returned with a vengeance!

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See the Chicago Board Options Exchange’s website, www.cboe.com/.

TRADED DERIVATIVE SECURITIES

Two Economic Motives Regulatory changes have powerful impacts on markets. In fact, economics Nobel laureate Merton Miller argued in a 1986 article that regulations and taxes cause financial innovation. The reason is because derivative securities are often created to circumvent government regulations that prohibit otherwise lucrative transactions. And because most countries tax income from different sources (and uses) at different rates, financial innovations are often designed to save tax dollars as well. The desire to lower transactions costs also influences financial innovation. This perspective comes from another Nobel Prize–winning University of Chicago economist, Ronald Coase. Financial institutions often devise derivatives so that brokerage commissions, the difference between a securities dealer’s buying and selling prices, are minimized (see Extension 1.1). This leads us to an (almost) axiomatic truth that will guide us throughout the book: trading moves to those markets where transaction costs and regulatory constraints are minimized.

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1.3

Traded Derivative Securities

A security, such as a stock or bond, gives its holder ownership rights over some assets and cash flows. It exists as a paper document or an electronic entry record and usually trades in an organized market. You may be familiar with bonds, which are debts of the issuer, and stocks, which give investors equity in the issuing company. Bonds and stocks require an initial investment. Most bonds pay back a promised amount (principal or par value) at maturity. Some bonds pay interest (coupons) on a regular basis, typically semiannually, while others are zero-coupon bonds that pay no interest but are sold at a discount from the principal. The investment in stock is never repaid. Stockholders usually get paid dividends on a quarterly basis as compensation for their stock ownership. Stock prices can increase and create capital gains for investors, and this profit is realized by selling the stock. Alternatively, stock prices can decrease and create capital losses. Stocks and bonds are often used to create new classes of securities called derivatives, and that’s where the variety comes in. As previously noted, a derivative security is a financial contract whose value is derived from one or more underlying assets or indexes—a stock, a bond, a commodity, a foreign currency, an index, an interest rate, or even another derivative security. Forwards, futures, and options are the basic types of derivatives. These are explained later in the book. Some common terminology will help us understand the various derivative contracts traded: 1. Real assets include land, buildings, machines, and commodities, whereas financial assets include stocks, bonds, and currencies—both real and financial assets have tangible values. 2. Notional variables include interest rates, inflation rates, and security indexes, which exist as notions rather than as tangible assets.

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EXTENSION 1.1: The Influence of Regulations, Taxes, and Transaction Costs on Financial Innovation In the old days, finance mainly consisted of legal issues, institutional description, and investment rules of thumb. This changed in the middle of the twentieth century, when financial economics sprang to life as an offshoot of economics. In a little over two decades, a new finance based on a rigorous analytics emerged. James Tobin and Harry Markowitz’s portfolio theory (late 1940s and early 1950s); Franco Modigliani and Merton Miller’s M&M propositions concerning the irrelevance of a firm’s capital structure and dividend policy (late 1950s and early 1960s); William Sharpe, John Lintner, and Jan Mossin’s capital asset pricing model (mid-1960s) and Fischer Black, Myron Scholes, and Robert Merton’s option pricing model (early 1970s) established the basic theories. All these works have been celebrated with the Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel, popularly known as the Nobel Prize in Economics. Other Nobel laureates, including Kenneth Arrow, Ronald Coase, Gerard Debreu, John Hicks, Paul Samuelson, and a well known economist, John Maynard Keynes, also contributed to finance. Two of these Nobel laureates’ views concerned financial innovation.

Miller’s View: Regulations and Taxes Spur Financial Innovation

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What leads to revolutionary change in financial institutions and instruments? University of Chicago professor Merton Miller (1923–2000) argued that a major cause of financial innovation is regulations and taxes. Innovative financial securities are often created to avoid government regulations that prohibit otherwise lucrative transactions. And because most countries tax income from different sources (and uses) at different rates, derivative securities can be designed to save tax dollars. Miller (1986) gives several examples: ■

Regulation Q of the US placed a ceiling on the interest rate that commercial banks could pay on time deposits. Although this wasn’t a problem during much of the postwar period, US interest rates rose above this ceiling during the late 1960s and early 1970s. When this occurred, US banks started losing customers. US banks realized, however, that Regulation Q did not apply to dollar-denominated time deposits in their overseas branches, and they soon began offering attractive rates via Eurodollar accounts. Interestingly, Regulation Q has long been repealed, but the Eurodollar market still continues to flourish.

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In the late 1960s, the US government imposed a 30 percent withholding tax on interest payments to bonds sold in the US to overseas investors. Consequently, for non-US citizens, the market for dollar-denominated bonds moved overseas to London and other money centers on the continent. This created the Eurobond market that still continues to grow today.

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The British government restricted dollar financing by British firms and sterling financing by non-British firms. Swaps, transactions in which counterparties exchange one form of cash flow for another, were developed to circumvent these restrictions.

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It was found in 1981 that US tax laws allowed a linear approximation for computing the implicit interest for long-term deep discount zero-coupon bonds. This inflated the present value of the interest deductions so much that a taxable corporation could actually profit by issuing a zero-coupon bond and giving it away! Not surprisingly, US corporations started issuing zero-coupon bonds in large numbers. This supply dwindled after the US Treasury fixed this blunder.

TRADED DERIVATIVE SECURITIES

9

Coase’s View: Transaction Costs are a Key Determinant of Economic Activity

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In 1937, Ronald Coase, a twenty-seven-year-old faculty member at the London School of Economics, published a simple but profound article titled “The Nature of the Firm.” Coase argued that transactions incur costs, which come from “negotiations to be undertaken, contracts have to be drawn up, inspections have to be made, arrangements have to be made to settle disputes, and so on,” and firms often appear when they can lower these transaction costs. With respect to financial markets, this logic implies that market participants often trade where they can achieve their objectives at minimum cost. Financial derivatives are often created so that these costs are minimized as well. The lowering of transaction costs as an economic motivation was especially important during the 1990s and the new millennium. Changes in the economic and political landscape and the IT revolution have made it possible to significantly lower transaction costs, even eliminating age-old professions like brokers and dealers from many trading processes. For example, this motive was a major reason why traders migrated from Treasury securities to Eurodollar markets. Eurodollar markets, being free from Fed regulations and the peculiarities of the Treasury security auction cycle, have fewer market imperfections and lower liquidity costs.

Writing financial contracts on values or cash flows determined by future realizations of real asset prices, financial asset prices, or notional variables creates a derivative security. Early derivatives were created solely from financial and real assets. For example, in the 1960s, agricultural commodity–based futures were the most actively traded derivative contracts in the US. However, as the economy has evolved, notional variables and derivative securities based on these notional variables have also been created. Notional variables are often introduced to help summarize the state of the economy. Just open the Wall Street Journal, and you will be amazed by the variety of indexes out there. You will find not only regular stock price indexes, such as the Dow-Jones and Standard and Poor’s, but a whole range of other indexes, including those based on technology stocks, pharmaceuticals stocks, Mexican stocks, utility stocks, bonds, and interest rates. In addition, notional values are often useful for the creation of various derivatives. Perhaps the most famous example of this is a plain vanilla interest rate swap whose underlyings are floating and fixed interest rates.

Diverse Views on Derivatives It is easy to speculate with derivatives: buy them, take huge one-sided bets, and laugh on the way to the bank or cry on the way to bankruptcy! Imprudent risk taking using derivatives is not uncommon, as illustrated by huge losses at many institutions, including Procter and Gamble (P&G) and Barings Bank. People are uncomfortable with derivatives because they are complex instruments: difficult to understand, highly leveraged, and oftentimes not backed by sufficient collateral. High leverage means that small changes in the underlying security’s price can cause large swings in the derivative’s value.

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For these reasons, derivatives attract strong views from both sides of the aisle. The renowned investors Warren Buffett and Peter Lynch dislike derivatives. In his “Chairman’s Letter” in Berkshire Hathaway’s 2002 annual report, Buffett characterized derivatives as “time bombs, both for the parties that deal in them and the economic system.” Lynch (1989) once stated that options and futures on stocks should be outlawed. These concerns were vindicated by the hundreds of billions of dollars of derivatives-related losses suffered by financial institutions during 2007 and 2008, which contributed to the severe economic downturn. By contrast, former Fed chairman Alan Greenspan opined in a speech delivered before the Futures Industry Association in 1999 that derivatives “unbundle” risks by carefully measuring and allocating them “to those investors most able and willing to take it,” a phenomenon that has contributed to a more efficient allocation of capital. And in Merton Miller on Derivatives, the Nobel laureate (Miller 1997, ix) assessed the impact of the “derivatives revolution” in glowing terms:

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Contrary to the widely held perception, derivatives have made the world a safer place, not a more dangerous one. They have made it possible for firms and institutions to deal efficiently and cost effectively with risks and hazards that have plagued them for decades, if not for centuries. True, some firms and some financial institutions have managed to lose substantial sums on derivatives, but some firms and institutions will always find ways to lose money. Good judgment and good luck cannot be taken for granted.

Nobel laureate physicist David Gross provides a nice analogy for this disagreement. He views knowledge as expanding outward like a growing sphere and ignorance as the surface of that sphere. With respect to derivatives, we have accumulated significant knowledge over the past thirty years, but with respect to the causes of tsunami-like financial crises, there is still much for us to learn.

Applications and Uses of Derivatives Derivatives trade in zero net supply markets, where each buyer has a matching seller. When hedging with a derivative, the other side of the transaction (counterparty) may be using it for speculative reasons. Hedging and speculation are often two sides of the same coin, and it is a zero-sum game because one trader’s gain is the other’s loss. Example 1.1 illustrates these concepts.

EXAMPLE 1.1: Hedging and Speculation in a Derivative Transaction ■

April is the beginning of the corn-growing season, the commodity used in our example.

■

Consider Mr. Short, a farmer in the midwestern United States. Short combines land, labor, seeds, fertilizers, and pesticides to produce cheap corn. He hopes to sell his corn harvest in September.

■

Expertise in growing corn does not provide a crystal ball for forecasting September corn prices. If Short likes sleeping peacefully at night, he may decide to lock in the selling price for September corn when he plants it in April.

TRADED DERIVATIVE SECURITIES

■

To see how this is done, let’s assume that everyone expects corn to be worth $10.00 per bushel in September. Ms. Long, a trader and speculator, offers to buy Short’s corn in September for $9.95 per bushel, which is the forward price.

■

To remove his risk, Short readily agrees to this forward price. Together they have created a forward contract—a promise to trade at a fixed forward price in the future. Although Short expects to lose 5 cents, he is happy to fix the selling price. The forward contract has removed output price uncertainty from his business. Short sees 5 cents as the insurance premium he pays for avoiding unfavorable future outcomes. Having hedged his corn selling price, Short can focus on what he knows best, which is growing corn.

■

Ms. Long is also happy—not that she is wild about taking risks, but she is rational and willing to accept some unwanted risk, expecting to earn 5 cents as compensation for this activity. Later in the book, you will see how a speculator may manage her risk by entering into another transaction at a better price but on the other side of the market.

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This simple example illustrates a key principle of historical importance—it was precisely to help farmers hedge grain prices that the first modern derivatives exchange, the CBOT, was established in 1848. Even the bitterest critics of derivatives have acknowledged this beneficial role. Nowadays, the uses of derivatives are many. Consider the following possibilities: ■

Hedging output price risk: A gold-mining company can fix the selling price of gold by selling gold futures.

■

Hedging input cost: A sausage maker can hedge input prices by buying pork belly futures.

■

Hedging currency risk: An American manufacturer buying machines from Germany for which the payment is due in three months can remove price risk from the dollar–euro exchange rates by buying a currency forward contract.

■

Hedging interest rate risk: A pension fund manager who is worried that her bond portfolio will get clobbered by rising interest rates can use Eurodollar futures (or options on those futures) to protect her portfolio.

■

Protecting a portfolio against a market meltdown: A money manager whose portfolio has reaped huge gains can protect these gains by buying put options.

■

Avoiding market restrictions: A trader can avoid short selling restrictions that an exchange might impose by taking a sell position in the options or the futures market.

■

Ruining oneself: A wretch can gamble away his inheritance with derivatives.

Before you finish this book, you will have a better understanding of the role derivatives play in each of these scenarios, except for the last one. The proclivity to gamble and lose arises from the depths of the human psyche, of which we have no special understanding.

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A Quest for Better Models Great recognition for derivatives came in 1997, when Robert Merton and Myron Scholes won the Nobel Prize in Economics for developing the Black–Scholes– Merton (BSM) model for option valuation. A co-originator of the model, Fischer Black, died earlier and missed receiving the prize, which is not given posthumously. This formula has become a staple of modern option pricing. Although extremely useful—options traders have it programmed into their computers—the BSM model makes a number of restrictive assumptions. These restrictive assumptions limit the application of the model to particular derivatives, those for which interest rate risk is not relevant (the BSM assumptions are discussed in later chapters). Subsequent researchers have relaxed these restrictive assumptions and developed pricing models suitable for derivatives with interest rate risk. The standard model used on Wall Street for this application is the Heath–Jarrow–Morton (HJM) model. The HJM model can be used to price interest rate derivatives, long-lived financial contracts, and credit derivatives. Credit derivatives are among the newest class of derivatives to hit Wall Street. Started in the early 1990s, they now compose a huge market, and they are the subject of intense public debate following the 2007 credit crisis and their role in it. Why develop and study better pricing models? Suppose you trade a variety of securities many times a day. If your models are better, you have an edge over your competitors in determining fair value. Over many transactions, when trading based on fair value—selling above fair value and buying below—your edge will pay off. Investment banking and Wall Street firms have no choice but to use the best models. It’s just too important for their survival. Powerful models also help traders manage risk.

1.4

Defining, Measuring, and Managing Risk

Risk is an elusive concept and hard to define. In finance and business, risk can be defined and measured in many ways, none of them completely universal. Risk often depends on the user’s perspective. We begin by looking at risk through a regulator’s eyes. Next, we discuss risk from an individual’s as well as an institutional trader’s viewpoint. Finally, we discuss risk from a corporate management perspective.

1.5

The Regulator’s Classification of Risk

In 1994, the international banking and securities regulators issued guidelines for supervising the booming derivatives market. The Basel Committee on Banking Supervision and the International Organization of Securities Commissions (IOSCO) recommended that for a stable world financial system, the national regulators had to

THE REGULATOR’S CLASSIFICATION OF RISK

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ensure that banks and securities firms have adequate controls over the risks they incur when trading derivatives.2 The Basel Committee’s Risk Management Guidelines for Derivatives (July 1994, 10–17) identified the following risks in connection with an institution’s derivative activities. IOSCO’s Technical Committee also issued a similar paper at the time:3 ■

Credit risk (including settlement risk) is the risk that a counterparty will fail to perform on an obligation.

■

Market risk is the risk to an institution’s financial condition resulting from adverse movements in the level or volatility of market prices. This is the same as price risk.

■

Liquidity risk in derivative activities can be of two types: one related to specific products or markets and the other related to the general funding of the institution’s derivative activities. The former is the risk that an institution may not be able to, or cannot easily, unwind or offset a particular position at or near the previous market price because of inadequate market depth or disruptions in the marketplace. Funding liquidity risk is the risk that the institution will be unable to meet its payment obligations on settlement dates or in the event of margin calls (which, we explain later, is equivalent to coming up with more security deposits).

■

Operational risk (also known as operations risk) is the risk that deficiencies in information systems or internal controls will result in unexpected loss. This risk is associated with human error, system failures, and inadequate procedures and controls.

■

Legal risk is the risk that contracts are not legally enforceable or documented correctly.

Managing market or price risk is the subject of this book. Initially, this topic attracted the sole attention of academics and practitioners alike. It still remains the most important risk for us to understand and to manage. The other risks only appear when normal market activity ceases. Credit risk evaluation is currently a subject of advanced research. As we explain later, exchange-traded derivatives markets are so designed that they are nearly free from credit risk. Liquidity risk is a persistent problem for traders who choose markets in which securities are not easily bought and sold. Chapter 3 discusses what makes markets illiquid. Operational risk is a reality with which one has to live. It is part of running a business, and appropriate management checks and balances reduce it. Legal risk isn’t a problem for exchange-

2 In 1974, the Group of Ten countries’ central-bank governors established the Basel Committee on Banking Supervision. The Basel (or Basle) Committee formulates broad supervisory standards and recommends statements of best practice in the expectation that individual authorities will implement (see “History of the Basel Committee and Its Membership,” www.bis.org/bcbs/history.htm). The International Organization of Securities Commissions, which originated as an inter-American regional association in 1974, was restructured as IOSCO in 1983. It has evolved into a truly international cooperative body of securities regulators (see www.iosco.org/about/about). 3

For both reports, see www.bis.org/publ. These definitions are modified only slightly from those contained in the referenced report.

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traded contracts; however, it’s a genuine problem in OTC markets. We leave this topic for the courts and law schools. This jargon-laden Basel Committee report also cited the need for appropriate oversight of derivatives trading operations by boards of directors and senior management and the need for comprehensive internal control and audit procedures. It urged national regulators to ensure that firms and banks operate on a basis of prudent risk limits, sound measurement procedures and information systems, continuous risk monitoring, and frequent management reporting. The Basel Committee reports have been extremely influential in terms of their impact on derivatives regulation. We return to the issues again in the last chapter of the book, after mastering the basics of derivatives securities.

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1.6

Portfolio Risk Management

Let us examine some issues in connection with portfolio risk management from the perspective of an individual or institutional trader. One of the first pearls of wisdom learned in finance is that to earn higher expected returns, one has to accept higher risks. This maxim applies to the risk of a portfolio, called portfolio risk. This portfolio risk, when applied to a single security, can be broken into two parts: (1) nondiversifiable risk, which comes from market-wide sources, and (2) diversifiable risk, which is unique to the security and can be eliminated via diversification. Pioneering works by Markowitz (1952), Sharpe (1964), Lintner (1965), and Mossin (1966) gave us the capital asset pricing model—the alphas, the betas—ideas and concepts used by market professionals and finance academics. Modern portfolio theory encourages investors to construct a portfolio in a topdown fashion. This involves three steps: (1) do an asset allocation, which means deciding how to spread your investment across broad asset classes such as cash, bonds, stocks, and derivatives; (2) do security selection, which means deciding which securities to hold within each asset class; and (3) periodically revisiting these issues, rebalancing and hedging the portfolio with derivatives as appropriate. Portfolio risk management is critical for investment companies such as hedge funds and mutual funds. These financial intermediaries receive money from the public, invest them in various financial securities, and pass on the gains and losses to investors after deducting expenses and fees.

1.7

Corporate Financial Risk Management

Risks That Businesses Face To understand the risks that businesses face, let’s take a look at a typical company’s balance sheet, a vital accounting tool that gives a snapshot of its financial condition by summarizing its assets, liabilities, and ownership equity on a specific date. An asset provides economic benefits to the firm, whereas a liability is an obligation that requires payments at some future date. The difference between the two accrues

CORPORATE FINANCIAL RISK MANAGEMENT

TABLE 1.1: Risks That a Business Faces Assets Current assets ■ Cash and cash equivalents (interest rate risk)

■

Accounts payable (interest rate risk, currency risk)

Accounts receivable (interest rate risk, currency risk)

■

Financial liabilities (interest rate risk, currency risk)

Inventories (commodity price risk) Long-term assets ■ Financial assets (interest rate risk, market risk, currency risk)

■

■

■

■

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Liabilities

Pension fund obligations (interest rate risk, market risk) Equity ■ Ownership shares

Property, plant, and equipment (interest rate risk, commodity price risk)

to the owners of the company as shareholder’s equity (hence the identity assets equals liabilities plus shareholder’s equity). Depending on whether they are for the short term (one year or less) or for the long haul, assets and liabilities are further classified as current or noncurrent, respectively. Table 1.1 illustrates some risks that affect different parts of a firm’s balance sheet. We see that a typical company faces three kinds of risks: currency risk, interest rate risk, and commodity price risk. These are the three components of market risk: 1. If a big chunk of your business involves imports and exports or if you have overseas operations that send back profits, then exchange rate risk can help or hurt. This is a risk that you must understand and decide whether to hedge using currency derivatives. 2. No less important is interest rate risk. It is hard to find companies like Microsoft, with tens of billions of dollars in cash holdings that can be quickly deployed for value-enhancing investments. Most companies are cash strapped. Interest rate fluctuations therefore affect their cost of funds and influence their investment activities. Interest rate derivatives offer many choices for managing this risk. 3. Unless a financial company, most businesses are exposed to fluctuations in commodity prices. A rise in commodity prices raises the cost of buying inputs that may not always be passed on to customers. For example, if crude oil prices go up, so does the price of jet fuel and an airline’s fuel costs. Sometimes airlines levy a fuel surcharge, but it is unpopular and often rolled back. Another option is to hedge such risks by using oil-price derivatives.

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Nonhedged Risks Despite the development of many sophisticated derivatives useful for hedging (and speculation), some risks are difficult, if not impossible, to hedge. For example, it is very hard to hedge operational risk. Recall that operational risk is the risk of a loss owing to events such as human error, fraud, or faulty management. Although a bank can buy insurance to protect itself from losses due to fire, no insurance company will insure a bank against the risk that a trader presses the wrong computer button and enters the wrong bond trade. For examples of such operational risk losses, see Chapter 26. Other difficult or impossible to hedge risks include losses because of changes in commodity prices for which there is no futures contract trading. We will now take a bird’s-eye view of how a blue chip company uses derivatives for risk management.

Risk Management in a Blue Chip Company

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The SFAS (Statement of Financial Accounting Standards) No. 133, “Accounting for Derivative Instruments and Hedging Activities,” as amended, requires US companies to report all derivative instruments on the balance sheet at fair value and establishes criteria for designation and effectiveness of hedging relationships. Let’s take a look at Form 10-K (annual report) filed with the US Securities and Exchange Commission by Procter & Gamble (P&G), a giant company that owns some of the world’s best-known consumer product brands.4 P&G heavily uses derivatives for risk management. A careful reading of this annual report reveals several interesting characteristics of P&G’s risk management activity: ■

P&G is exposed to the three categories of risk that we just mentioned.

■

P&G consolidates these risks and tries to offset them naturally, which means some risks cancel each other. It then tries to hedge the rest with derivatives.

■

P&G does not hold derivatives for trading purposes.

■

P&G monitors derivatives positions using techniques such as market value, sensitivity analysis, and value at risk. When data are unavailable, P&G uses reasonable proxies for estimating volatility and correlations of market factors.

■

P&G uses interest rate swaps to hedge its underlying debt obligations and enters into certain currency interest rate swaps to hedge the company’s foreign net investments.

■

P&G manufactures and sells its products in many countries. It mainly uses forwards and options to reduce the risk that the company’s financial position will be adversely affected by short-term changes in exchange rates (corporate policy limits how much it can hedge).

■

P&G sometimes uses futures, options, and swaps to manage the price volatility of raw materials.

4

See P&G’s 2015 Annual Report (http://www.pginvestor.com/CustomPage/Index?KeyGenPage=1073748359)

RISK MANAGEMENT PERSPECTIVES IN THIS BOOK

■

P&G designates a security as a hedge of a specific underlying exposure and monitors its effectiveness in an ongoing manner.

■

P&G’s overall currency and interest rate exposures are such that the company is 95 percent confident that fluctuations in these variables (provided they don’t deviate from their historical behavior) would not materially affect its financial statements. P&G expects significant risk neither from commodity hedging activity nor from credit risk exposure.

■

P&G grants stock options and restricted stock awards to key managers and directors (employee stock options are valued by using a binomial model that you will learn to implement in Part III of this book).

How does one understand, formulate, and implement hedging and risk management strategies like the ones adopted by P&G? Read on, for that is a major purpose of this book.

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1.8

Risk Management Perspectives in This Book

The Basel Committee’s risk nomenclature has become the standard way that market participants see, classify, and understand different types of risks in derivatives markets. In this book, we restrict our focus mainly to market or price risk. For this we usually wear four separate hats and look at risk management from the perspectives of an individual trader, a financial institution, a nonfinancial corporation, and a dealer. In early chapters, we start by examining financial securities and their markets from an individual trader’s standpoint. This simple and intuitive approach makes it easier to understand the material. It’s also historically correct because many markets, including those for stocks, futures, and options, were started by individual traders. But today’s derivatives markets have become playing fields for financial institutions. This is no surprise because as an economy develops, a greater share of its gross domestic product comes from services, of which financial institutions are a major constituent. These institutions generally engage in sophisticated ways of investing. We discuss derivatives and risk management from a financial institution’s viewpoint because you may eventually be working for one, or at the very least, you are likely to have financial dealings with such institutions on a regular basis. The term institution is a catchall phrase that includes commercial and investment banks, insurance companies, pension funds, foundations, and finance companies such as mutual funds and hedge funds. In later chapters, we study swaps and interest rate derivatives, whose markets are the near-exclusive domain of financial (and nonfinancial) institutions. Nonfinancial companies engage in more real activity than financial companies. Nonfinancials give us food, develop medicines, build homes, manufactures cars, refine crude oil to create gasoline, generate electricity, provide air travel, create household chemicals, and make computers to save and express our ideas. They buy one or more inputs to produce one or more outputs, and different kinds of risk (including exchange rate risk, interest rate risk, and commodity price risk) can

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seriously affect their businesses. This takes us to financial engineering, which applies engineering tools to develop financial contracts to meet the needs of an enterprise. This is our third hat: looking at derivatives from a nonfinancial company’s risk management perspective. Sometimes, we take a dealer’s perspective. A dealer is a financial intermediary who posts prices at which she can buy (wholesale or bid price) or sell (retail or ask price) securities to her customers. Trying to make a living from the spread, or the difference between these two prices, the dealer focuses on managing books, which means carefully controlling her inventory of securities to minimize risk. Both an individual trader and a financial institution can play a dealer’s role in financial markets. These four perspectives aren’t ironclad, and we flit from one to another as the discussion demands. In the final analysis, this shifting from one category to another isn’t bad for the aspiring derivatives expert. In the words of the immortal bard William Shakespeare, from Hamlet (if we take the liberty of forgetting about spirits and instead apply this to the mundane), “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” So stay awake and study derivatives, develop a sense for risky situations, understand the markets, learn pricing models, and know their limitations.

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1.9

Summary

1. Derivatives are financial securities that derive their value from some underlying asset price or index. Derivatives are often introduced to hedge risks caused by increasingly volatile asset prices. For example, during the last three decades of the twentieth century, we have moved away from a regime of fixed exchange rates to a world of floating exchange rates. The market-determined foreign exchange rates increased their volatility, creating the need for foreign currency derivatives. 2. In today’s interconnected global economy, risks coming from many different sources can make or break businesses. Derivatives can both magnify risk (leverage and gambling) or reduce risk (hedging). Although gambling with derivatives remains popular in some circles, and derivatives mishaps grab newspaper headlines, most traders prudently use derivatives to remove unwanted risks affecting their businesses. 3. There are many types of risk. Besides market risk, the regulators are concerned about credit risk, liquidity risk, operational risk, and legal risk. Though unglamorous, these risks can eat away profits and jolt the running of smooth-functioning derivatives markets. 4. Derivatives are very useful for managing the risk of a portfolio, which is a collection of securities. Portfolio risk management is important for both individual investors and financial companies such as hedge funds and mutual funds. Many businesses are exposed to exchange rate risk, interest rate risk, and commodity price risk and use derivatives to hedge them. US companies report derivatives usage and exposure on Form 10-K filed annually with the Securities and Exchange Commission. For example, P&G reports that it is exposed to exchange rate risk,

QUESTIONS AND PROBLEMS

interest rate risk, commodity price risk, and credit risk. P&G consolidates risks and attempts to offset them naturally and tries to hedge the remaining risk with derivatives. 5. We look at risk management from several different perspectives: that of an individual trader, a financial institution, a nonfinancial corporation, and a dealer. These perspectives aren’t ironclad, and we flit from one to another as the discussion demands.

1.10

Cases

Hamilton Financial Investments: A Franchise Built on Trust (Harvard Business

School Case 198089-PDF-ENG). The case discusses various risks faced by a finance company that manages mutual funds and provides discount brokerage services. Grosvenor Group Ltd. (Harvard Business School Case 207064-PDF-ENG). The

case considers whether a global real estate investment firm should enter into a property derivative transaction to alter its asset allocation and manage its business. Societe Generally (A and B): The Jerome Kerviel Affair (Harvard Business

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School Cases 110029 and 110030-PDF-ENG). The case illustrates the importance of internal control systems in a business environment that involves a high degree of risk and complexity in the context of a derivatives trader indulging in massive directional trades that went undetected for over a year.

1.11

Questions and Problems

1.1. What is a derivative security? Give an example of a derivative and explain why

it is a derivative. 1.2. List some major applications of derivatives. 1.3. Evaluate the following statement: “Hedging and speculation go hand in hand

in the derivatives market.” 1.4. What risks does a business face? 1.5. Explain why financial futures have replaced agricultural futures as the most

actively traded contracts. 1.6. Explain why derivatives are zero-sum games. 1.7. Explain why all risks cannot be hedged. Give an example of a risk that cannot

be hedged. 1.8. What is a notional variable, and how does it differ from an asset’s price? 1.9. Explain how derivatives give traders high leverage. 1.10. Explain the essence of Merton Miller’s argument explaining what spurs

financial innovation.

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1.11. Explain the essence of Ronald Coase’s argument explaining what spurs

financial innovation. 1.12. Does more volatility in a market lead to more use of financial derivatives?

Explain your answer. 1.13. When the international banking regulators defined risk in their 1994 report,

what definition of risk did they have in mind? How does this compare with the definition of risk from modern portfolio theory? 1.14. What’s the difference between real and financial assets? 1.15. Explain the differences between market risk, credit risk, liquidity risk, and

operational risk. 1.16. Briefly present Warren Buffett’s and Alan Greenspan’s views on derivatives. 1.17. Consider the situation in sunny Southern California in 2005, where house

prices have skyrocketed over the last few years and are at an all-time high. Nathan, a software engineer, buys a second home for $1.5 million. Five years back, he bought his first home in the same region for $350,000 and financed it with a thirty-year mortgage. He has paid off $150,000 of the first loan. His first home is currently worth $900,000. Nathan plans to rent out his first home and move into the second. Is Nathan speculating or hedging? 1.18. During the early years of the new millennium, many economists described the

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past few decades as the period of the Great Moderation. For example, ■

an empirical study by economists Olivier Blanchard and John Simon found that “the variability of quarterly growth in real output (as measured by its standard deviation) had declined by half since the mid-1980s, while the variability of quarterly inflation had declined by about two thirds.”

■

an article titled “Upheavals Show End of Volatility Is Just a Myth” in the Wall Street Journal, dated March 19, 2008, observed that an important measure of stock market volatility, “the Chicago Board Options Exchange’s volatility index, had plunged about 75% since October 2002, the end of the latest bear market, through early 2007”; the article also noted that “in the past 25 years, the economy has spent only 16 months in recession, compared with more than 60 months for the previous quarter century.” a. What were the explanations given for the Great Moderation? b. Does the experience of the US economy during January 2007 to Decem-

ber 2010 still justify characterizing this as a period of Great Moderation? Report (1) quarterly values for changes in the gross domestic product, (2) quarterly values for changes in the inflation rate, and (3) the volatility VIX Index value during this period to support your answer. 1.19. Drawing on your experience, give examples of two risks that one can easily

hedge and two risks that one cannot hedge.

QUESTIONS AND PROBLEMS

1.20. Download Form 10-K filed by P&G from the company’s website or the

US Securities and Exchange Commission’s website. Answer the following questions based on a study of this report: a. What are the different kinds of risks to which P&G is exposed? b. How does P&G manage its risks? Identify and state the use of some

derivatives in this regard. c. Name some techniques that P&G employs for risk management. d. Does P&G grant employee stock options? If so, briefly discuss this program.

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What valuation model does the company use for valuing employee stock options?

21

2 Interest Rates 2.1 Introduction 2.2 Rate of Return

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2.3 Basic Interest Rates: Simple, Compound, and Continuously Compounded EXTENSION 2.1 Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time

2.4 Discounting (PV) and Compounding (FV): Moving Money across Time EXTENSION 2.2 Moving Multiple Cash Flows across Time

2.5 US Treasury Securities 2.6 US Federal Debt Auction Markets

2.7 Different Ways of Investing in Treasury Securities The Treasury Auction and Its Associated Markets The Repo and the Reverse Repo Market EXTENSION 2.3 A Repurchase Agreement Interest Rate Derivatives

2.8 Treasury Bills, Notes, Bonds, and STRIPS 2.9 Libor versus a Libor Rate Index 2.10 Summary 2.11 Cases 2.12 Questions and Problems

INTRODUCTION

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2.1

Introduction

Gentlemen prefer bonds. Perhaps you are chuckling at our “misquotation” of the famous Marilyn Monroe movie title Gentlemen Prefer Blondes. In fact, former US treasury secretary Andrew Mellon made this observation about bonds many decades before the movie.1 In all likelihood, he said this because bonds are considered safer than stocks. Corporations may miss dividend payments on stocks, but bonds are legally bound to pay promised interest on fixed dates. If the interest payments are not paid, default occurs, and the corporations are vulnerable to lawsuits and bankruptcy. However, at this stage in our presentation, we will assume that all bonds considered have no default (credit) risk. This greatly simplifies the analysis. Consequently, the bonds considered in this chapter are best viewed as US Treasury securities. Credit risk is considered only later in the book (in Chapter 26), after we have mastered default-free securities. Most bonds make interest payments according to a fixed schedule. Hence bonds are also called fixed-income securities. Bonds are useful for moving money from one period to another. Does money retain its value over time? Not really. Inflation, the phenomenon of a rise in the general price level, chips away money’s buying power little by little, year by year. A popular measure of the US inflation rate is the change in the consumer price index (CPI). At each date, the CPI measures the price of a fixed bundle of nearly two hundred goods and services that a typical US resident consumes. Unless you are buying computers and electronic gizmos whose prices have drastically fallen because of technological advances, a hundred dollar bill doesn’t quite buy as much as it did ten years ago. To reduce the cost of inflation, money earns interest. Instead of stashing cash under a mattress, one can put it into a savings account and see it grow safely. What, then, is an interest rate? An interest rate is the rate of return earned on money borrowed or lent. Ponder and you realize that the cost of borrowing a hundred dollar bill is the interest of perhaps $5 per year that’s paid to the lender. Suppose that the inflation rate is 3 percent per year, which is close to the US inflation rate in the new millennium. If the interest rate for risk-free loans is 3.10 percent per year, then borrowing is incredibly cheap, while 30 percent for such loans would be awfully expensive. Interest rates partly compensate the lender for inflation and partly reward her for postponing consumption until a later date. There are many ways of computing interest rates. Simple, compound, and continuously compounded rates are the three basic kinds. Each interest rate concept has its own use in finance theory as well as in practice. There are interest rates for risky as well as riskless loans. The first interest rates studied in this book are risk-free rates. How do you find risk-free interest rates? United States Government Treasury securities (the “Treasuries”) are considered default-free because of the taxing authority of the mighty federal government. Risk-free interest rates of various maturities can be easily extracted from their prices. We briefly explain how to do this extraction. Then we explain how the US Treasury securities market works, and we describe the 1

Andrew William Mellon (1855–1937) was US secretary of the treasury under three presidents (1921– 32). Source: US Treasury’s website, www.ustreas.gov/education/history/secretaries/awmellon.html.

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bills, notes, bonds, STRIPS, and TIPS that trade in these markets. We conclude the chapter with a discussion of the London Interbank Offered Rate (libor) and a liborbased rate index, which is a key rate used in the global interest rate derivatives market. We begin by introducing the concept of a rate of return. This, in turn, will lead us to a discussion of the three basic types of interest rates.

2.2

Rate of Return

In finance, we constantly talk about rates of return computed from prices. Rates of return measure how much an investment has earned. For example, if you buy a security for $50 today and sell it for $52 after six months, then you have earned $2 on your investment. The rate of return earned on your investment over this time period is 52 − 50 = 0.04, or 4 percent (when multiplied by 100 to express as a percentage) 50 It is hard to compare rates of return unless we explicitly mention the time interval over which they are computed. We need to know whether we are computing these over six months, or thirty-seven weeks, or ninety-one days. Consequently, it is worthwhile to standardize by computing them over a year. In this case, the annualized rate of return is

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12 ( 6 ) × 0.04 = 0.08, or 8 percent per year A more precise value for the rate of return may be obtained by considering the actual number of days for which the money was invested using 365 days in a year (some computations use 366 days in case of a leap year). Assuming that there were 181 days in this six-month period, we find the (annualized) rate of return as 365 ( 181 ) × 0.04 = 0.0807, or 8.07 percent per year The capital gain, the difference between the selling (final) and buying (initial) price, is the $2 earned on this investment. But there could be other cash flows at intermediate dates. For example, bonds make coupon payments and stocks pay dividends. If our $50 investment also received $0.50 interest at some intermediate date, then the (annualized) rate of return is 52 + 0.50 − 50 365 ( 181 ) × ( ) = 0.1008, or 10.08 percent per year 50 A profit of $2.50 is made on this investment. Expenses like brokerage account fees can cut into your profit. A profit occurs when there’s an overall gain from an investment or business activity. Businesses define it as total revenue minus total costs and record a profit if this is a positive number and a loss if it’s negative. In finance, we define Profit or loss = (Selling price + Income) − (Buying price + Expenses)

(2.1)

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

where the selling price is the final price, the buying price is the initial price, and a profit or loss happens if the result is positive or negative, respectively.2 We generalize this to develop a formula for computing the annualized rate of return (which is also called arithmetic return).

RESULT 2.1 Annualized Rate of Return The annualized arithmetic rate of return is 365 Rate of return = ( T ) ×

Selling price + Income − Expenses − Buying price ) ( Buying price

=(

Profit or loss 365 × ) ( Buying price ) T

(2.2)

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where T is the time interval, measured by the number of days over which the investment is held, and Income and Expenses denote positive and negative cash flows, respectively, from the investment (Rate of return is often multiplied by 100 and expressed as a percentage).

We need some reasonable conventions regarding rounding and reporting numbers and for measuring time, which we use throughout the book. They are discussed in Extension 2.1.

2.3

Basic Interest Rates: Simple, Compound, and Continuously Compounded

An interest rate is the rate of return promised (in case the money is yet to be lent) or realized (in case the money has already been lent) on loans. There seem to be a zillion ways of computing interest. Financial institutions like banks and credit card companies have devised clever ways of charging interest that appear less than they really are. That’s why many governments came up with a standard yardstick to help customers understand interest costs. For example, in the United States, all lenders have to disclose the rate they are charging on loans in terms of an Annual Percentage Rate (APR), a rate that annualizes using a simple interest rate. 2

An exception is short selling (see Chapter 3), in which selling occurs first and buying second, but the expression retains the positive and negative signs associated with these expressions.

25

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CHAPTER 2: INTEREST RATES

EXTENSION 2.1: Conventions and Rules for Rounding, Reporting Numbers, and Measuring Time To maintain consistency and to minimize unwanted errors, we follow some rules for rounding, reporting numbers, and calculations. We usually round to four places after the decimal point when reporting results. In all calculations, before reporting the numbers in the text, we retain 16 digits for accuracy. Any differences between reported results based on rounded numbers and calculated results (manipulations of the actual numbers) are due to these rounding errors: ■

If the number in the fifth place is more than 5, then add 1 to the fourth digit, e. g., 0.23456 becomes 0.2346.

■

If the number in the fifth place is less than 5, then keep it unchanged, e. g., 0.11223 becomes 0.1122.

■

If it’s exactly 5 and there are numbers after it, then add 1 in the fourth place, e. g., 0.123452 becomes 0.1235.

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In some contexts, we need to round to more places after the decimal point. The context will indicate when this is appropriate. Though we typically report numbers rounded to four places after the decimal point, the final dollar result is usually rounded to two places after the decimal. The dollar sign “$” is usually attached only in the final answer. We sometimes omit dollar signs from the prices when the context is understood. Different markets follow different time conventions. Treasury bill prices use the actual number of days to maturity under the assumption that there are 360 days in a year. Many banks pay interest compounded daily. Swaps use simple interest applied to a semiannual period, and many formulas use a continuously compounded rate. Most models measure time in years. If we start computing at time 0 and the security matures at time T, then it has a life of T years. If we start our clock at time t, then the life is (T – t) years. We prefer to use the first convention in our formulas. Time periods are converted by the following conventions: ■

If the time period is computed in days, then use the exact number of days and assume 365 days in the year (unless noted otherwise), e. g., 32 days will be 32/365 = 0.0877 year.

■

If the time period is computed in weeks, use fifty-two weeks in the year, e. g., seven weeks will be 7/52 = 0.1346 year.

■

If the time period is computed in months, use twelve months in the year, e. g., five months will be 5/12 = 0.4167 year. To summarize, we employ the following rules and conventions.

Rules of Rounding and Conventions of Reporting Numbers and Measuring Time ■

Reported numbers are rounded to four places after the decimal point. All calculations are performed using 16 digit numbers, not the rounded numbers. Any differences between reported results based on rounded numbers and calculated results (manipulations of the actual numbers) are due to these rounding errors.

■

The final dollar result is rounded to two places after the decimal with the dollar sign attached only in the final answer.

■

Time is measured in years. If we start computing from time 0 and the security matures at time T, then it has a life of T years; if the clock starts at time t, then the security has a life of (T – t) years.

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

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There are many different ways of computing interest rates. For a certain quoted rate, the realized interest can vary with (1) the method of compounding, (2) the frequency of compounding (yearly, monthly, daily, or continuously compounded), (3) the number of days in the year (52 weeks; 360 days, or 365, or 366 for a leap year), (4) the number of days of the loan (actual or in increments of months), and (5) other terms and conditions (collect half of the loan now, the other half later; keeping a compensating balance). Rather than bombard you with a list of interest rate conventions for different markets, we focus on the three basic methods: simple, compound, and continuously compounded interest rates. Understanding these concepts will lead to discounting and compounding, which will enable us to transfer cash flows across time. A simple interest rate is used in some sophisticated derivatives like caps and swaps. Money is not compounded under simple interest rates. An annual rate is quoted. When computed over several months, a fraction of the annual rate is used. Compound interest is when interest is earned on both the original principal and the accrued interest. Banks offer interest on daily balances kept in your account. When compounding interest, divide the annual rate by the number of compounding intervals (daily, weekly, monthly) and multiply these interest components together to compute the loan value at maturity. Continuous compounding pays interest on a continuous basis. One can view continuous compounding as the limit of compound interest when the number of compounding intervals gets very large and the time between earning interest gets very small! Example 2.1 demonstrates these three methods of interest computation.

EXAMPLE 2.1: Simple, Compound, and Continuously Compounded Interest Rates

Simple Interest Rates ■

Suppose you invest $100 today, and 6 percent is the simple interest rate per year. - A year from now, it will grow to $106. - After six months, it will grow to 100 × [1 + 0.06(1/2)] = $103. - After one day, it will be 100 × [1 + 0.06(1/365)] = 100.0164 = $100.02. - After 250 days, it will be 100 × [1 + 0.06(250/365)] = $104.11.

■

Summoning the power of algebra, we replace numbers with symbols to develop a formula for money growth under simple interest. Here L = $100 is the principal, which is the original amount invested. It is invested at the rate of interest i = 6 percent per year for T years, where T is measured in years or a fraction of a year. In the preceding expressions, T takes the values 1, 1/2, 1/365, and 250/365, respectively. Consequently, under simple interest, L dollars invested at i percent per year becomes after one year L(1 + i) and after T years L(1 + iT). This is illustrated in Figure 2.1. Notice that the simple interest rate is also the rate of return. This is verified by plugging the values in relation to Equation 2.2 to get (365/365)[L(1 + iT) − L]/L = i as T is one year.

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CHAPTER 2: INTEREST RATES

Compound Interest Rates ■

What about interest earning interest? This leads to the notion of compounding, which involves computing interest on both the principal and the accumulated interest. We split the time period T over which we are investing our money into m intervals. Within each interval, money earns simple interest. At the end of each interval, the principal and accumulated interest become the new principal on which interest is earned. If the interest is compounded semiannually, one year from now, $100 will grow to3 1 1 100 × [1 + 0.06 ( )] × [1 + 0.06 ( )] 2 2 1 2 = 100 × [1 + 0.06 ( )] 2 = $106.09 Compounding three times a year, one year from now, it will grow to 1 3 100 × [1 + 0.06 ( )] = $106.12 3 What about daily compounding that your local bank offers? With daily compounding, a year from now, the amount invested will grow to 100 × [1 + 0.06 (

1 1 × [1 + 0.06 ( … 365 times )] 365 365 )]

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= 100 × [1 + 0.06 (

365 1 365 )]

= $106.18 Now consider a loan for T years. Continuing with our example of daily compounding, after 250 days (or 250/365 year), $100 will grow to 100 × [1 + 0.06 (

1 1 × 1 + 0.06 ( … 250 times 365 )] [ 365 )]

= 100 × [1 + 0.06 (

250 1 365 )]

= $104.19 Rewriting the second line of the preceding expression with an eye toward generalization, we get 250

(365)( 1 365 ) 100 × [1 + 0.06 ( )] 365

(The book’s appendix gives the rules underlying these manipulations) ■

As before, we replace numbers with symbols to develop a formula for computing compound interest. Let L = $100 be the principal, which is invested at i = 6 percent per year for T years; however,

BASIC INTEREST RATES: SIMPLE, COMPOUND, AND CONTINUOUSLY COMPOUNDED

the interest is compounded m times every year. In the preceding expressions, m takes the values 2, 3, and 365, respectively. T = 1 year, except in the last example where it takes the value (250/365). Consequently, under compound interest, L dollars invested for one year becomes L(1 + i/m)m , and when invested for T years, it becomes (which is also shown in Figure 2.1) L (1 +

i mT m)

FIGURE 2.1: Simple, Compound, and Continuously Compounded Interest Rates

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Simple Interest (Interest does not earn interest) Time 0

1 year

L Investment

L(1 + i) Value after 1 year

Time T

L(1 + iT) Value at time T

Compound Interest (Interest earns interest) Time 0

Time 1/m

Time 2/m

1 year

L Investment

L(1 + i/m)

L(1 + i/m)2

L(1 + i/m)m Value after 1 year

Time T

L(1 + i/m)mT Value at time T

Continuously Compounded Interest (Interest earns interest continuously) Time 0

1 year

L Investment

Ler1 Value after 1 year

Time T

LerT Value at time T

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30

CHAPTER 2: INTEREST RATES

Continuously Compounded Interest Rates ■

But why stop at 365? Why not compound 1 million times a year? Then the investment grows to $106.1836545 at year’s end. Two things are happening here. As the compounding frequency increases, so does earned interest, but it grows at a decreasing rate—the money doesn’t grow very fast after a while. For example, compounding 10 million times will only give $106.1836547 at year’s end.

■

We can talk about “continuous compounding,” where the interest rate r (we now use r instead of i to denote the interest rate) is continuously compounded. This happens when the annual frequency of compounding m becomes larger and larger but the interest rate charged r/m becomes smaller and smaller during each of the compounding intervals. In that case, $1 becomes 1er at year’s end, where the exponential function e is the base of the natural logarithm (mathematically, we define log x ≡ ln (x) = M as equivalent to eM = x). It is approximated as 2.7183. Calculators have this as ex or exp ( … ), denoting the exponential function.

■

But that was for one year. The dollar investment grows to erT after T years. To summarize, under continuous compounding, L dollars invested at r percent per year for one year becomes Ler , and when invested for T years, it becomes LerT . (The appendix shows how this happens, and Figure 2.1 illustrates the result in a diagram.) In our example, this will give at year’s end 100e0.06 = $106.1836547

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Notice that the APR remains at 6 percent but the effective annual interest rate = er − 1 = 6.1837 percent. This is an example of a continuously compounded return (or logarithmic return). Continuously compounded interest rates are used as an input in the Black–Scholes–Merton model. Our example suggests that continuous compounding (which gives $106.18 at year’s end) is a much better approximation to daily compounding (which also gives $106.18) than is simple interest (which only gives $106). 3

The appendix discusses how to write expressions involving exponents (or indexes). It also gives rules for manipulating such expressions. Notice that the APR remains 6 percent, but your investment will earn more than this at year’s end. This is captured by the concept of an effective annual interest rate (EAR), which expresses the interest rate realized on a yearly basis. This is given by (1 + i/m)m - 1, where m is the frequency of compounding. Here, because m is 2, the EAR is 0.0609 or 6.09 percent.

We collect these three interest computation methods as Result 2.2.

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

RESULT 2.2 Simple, Compound, and Continuously Compounded Interest Rates Simple Interest A principal of L dollars when invested for T years becomes L (1 + iT)

(2.3)

where i percent per year is the simple interest rate.

Compound Interest L dollars invested for T years becomes L[1 + (i/m)]mT

(2.4)

where i percent per year is the compound interest rate and m is the number of times the interest is compounded every year.

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Continuously Compounded Interest L dollars invested for T years becomes LerT

(2.5)

where r percent per year is the continuously compounded interest rate and e is the exponential function.

2.4

Discounting (PV) and Compounding (FV): Moving Money across Time

We can use interest rates to find the dollar return or to price a zero-coupon bond. If we invest a dollar today and it continues to earn interest that we do not withdraw, then the final amount on the maturity date is the dollar return. The dollar return measures how an invested dollar grows over time. In real life, you can earn a dollar return by investing in a money market account (mma), which earns the risk-free rate in each period.

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CHAPTER 2: INTEREST RATES

The inverse of a dollar return is the price of a zero-coupon bond. A zero-coupon bond (or zero) sells at a discount, makes no interest (or coupon) payments over the bond’s life, and pays back the principal (or the par value) on the maturity date. A zero-coupon bond and a mma are closely related securities; the price of a zerocoupon bond (B) and the dollar return (1 + R) obtained from investing in a mma are inverses of one another, B ≡ 1/(1 + R). Both mmas and zeros help us move funds across time. Example 2.2 shows how to compute a dollar return and the price of a zero.

EXAMPLE 2.2: Computing a Dollar Return, Pricing a Zero, and Moving Funds across Time ■

Suppose that the simple interest rate is 6 percent per year. Today is time 0, and the bond matures at time T = 1/2 year. Then the dollar return after six months of investing in a mma is 1 + R = 1 × [1 + 0.06 (1/2)] = $1.03 This dollar return is the future value of $1 invested today.

■

Today’s price of a zero-coupon bond that pays $1 after six months is B ≡ 1/ (1 + R) = 1/1.03 = $0.9709 = $0.97

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This is the present value of $1 received after six months.

RESULT 2.3 Compounding and Discounting Cash Flows with Dollar Returns and Zero-Coupon Bonds Consider cash flows C(0) available today (time 0) and C(T) available at some later date (time T). Then multiplication of today’s C(0) by the dollar return (or equivalently, division by a zero-coupon bond price) gives the future value C (0) (1 + R) = C (0) /B

(2.6)

Multiplication of the future C(T) by a zero-coupon bond price (or division by the dollar return) gives the present value (or discounted value) BC (T) = C (T) / (1 + R)

(2.7)

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

where (1 + R) is the dollar return at time T from investing $1 today, B ≡ 1/(1 + R) is today’s price of a zero-coupon bond that pays $1 at time T, and T is the time to maturity in years. See Figure 2.2. In the case of simple interest, 1 + R = (1 + i × T)

(2.8)

where i is the simple interest rate per year. In the case of continuously compounded interest, 1 + R = erT

(2.9)

where r is the continuously compounded interest rate per year.

FIGURE 2.2: Compounding and Discounting Cash Flows using the Dollar Return (1 + R) or the Zero-Coupon Bond Price (B)

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Now (Time 0)

Future (Time T)

C(0)

C(0)(1 + R) = C(0)/B

BC(T) = C(T)/(1 + R)

C(T)

You can easily extend this result to move multiple cash flows across time, which may occur at different time periods (see Extension 2.2). Example 2.3 demonstrates

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CHAPTER 2: INTEREST RATES

how to compound and discount when a continuously compounded risk-free interest rate (r) is given.

EXAMPLE 2.3: Computing Present Values Using a Continuously Compounded Rate ■

Suppose that we receive $100 after six months. Let the continuously compounded risk-free interest rate r be 6 percent per year. Consequently, C(T) = $100, r = 0.06, and T = 0.5 year. Using expressions (2.4b) and (2.4d), the present value (PV) of $100 is 100 e−0.06 × 0.5 = 97.0446 = $97.04

■

Using the notation defined previously, B = $0.970446. If you invest this at a continuously compounded rate of 6 percent, you will get $1 in 6 months. Hence we can also write the PV of $100 in six months as 100B.

■

Notice that (1 + R) = 1/B = 1.0304545. Consequently, expressions (2.4a) and (2.4d) give the future value (FV) of $100 after six months as

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100 (1 + R) = $103.05

Continuous compounding will be useful in powerful options pricing models like the Black–Scholes–Merton model. If you are trying to compute options prices and you know the market price of a zero, then, as explained in Example 2.4, you can extract the risk-free continuously compounded rate and plug it into the Black– Scholes–Merton model.

EXTENSION 2.2: Moving Multiple Cash Flows across Time Moving cash flows across time is a fundamental tool that we use throughout the book. One can use interest rates, zero-coupon bonds, and mma values to do this.

EXT. 2.2 EX. 1: Moving Multiple Cash Flows across Time ■

Suppose you graduate a year from now and expect to get a job from which you squirrel away $5,000 at year’s end. Moreover, you expect a year-end bonus of $10,000. Assuming that we can commit “several sins”—(1) disregard the basic principle that risky cash flows must be discounted by risky discount rates and (2) operate under the assumption that the same interest rate applies to loans of different maturities—let us borrow against these future cash flows and use the funds to help pay your college tuition.

35

DISCOUNTING (PV) AND COMPOUNDING (FV): MOVING MONEY ACROSS TIME

■

Your bank offers you a loan at 6 percent interest, compounded daily. Approximating this by continuous compounding, we let r = 6 percent be the continuously compounded risk-free interest rate. Then the price of a zero-coupon bond maturing in t = 2 years is given by B(2) = 1/ert = e-rt = e-0.06 × 2 = $0.8869 (rounded to 4 decimal places). Writing C1 (2) = $5,000 and C2 (2) = $10,000, discounting the individual cash flows and adding them up gives B(2)C1 (2) = 0.8869 × 5,000 = $4,434.60 and B(2)C2 (2) = 0.8869 × 10,000 = $8,869.20, whose sum is B (2) C1 (2) + B (2) C2 (2) = $13, 303.81 (2.10)

■

Alternatively, you can add up the cash flows and then discount by multiplying by the zero-coupon bond price: B (2) [C1 (2) + C2 (2)] = 0.8869 × (5, 000 + 10, 000) = $13, 303.81

(2.11)

Both approaches give the same answer. In the actual calculations, B(1) and B(2) are not rounded to 4 decimal places (see Extension 2.1). ■

The equality of expressions (1) and (2) can be generalized to yield a formula for discounting cash flows belonging to a particular time period, t = 2: B (2) C1 (2) + B (2) C2 (2) = B (2) [C1 (2) + C2 (2)]

(2.12)

You can express this compactly by using the summation sign (∑). Then expression (3) can be written as 2

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B (2) ■

2

Cs (2) = B (2) Cs (2) [∑ ] ∑ s=1 s=1

(2.13)

What about extending this result to cash flows available at different time periods? Continuing with our previous example, suppose your grandparents have been saving for your college education by investing in a tax-favored education savings account. Your paternal grandparents will gift you C1 (1) = $12,000 and your mother’s parents will give you C2 (1) = $10,000 at time t = 1. Assuming that the interest rate r is 6 percent, and the price of a zero-coupon bond maturing in one year is B(1) = e-0.06 × 1 = $0.9418, the present value of these cash flows is B (1) [C1 (1) + C2 (1)] = 0.9418 × (12, 000 + 10, 000) = $20, 718.82

■

Suppose you want to determine how much of a loan you can take out today based on these future cash flows. For this you need to compute the present value of the four cash flows, two after one year and two after two years. They have the same value, $34,022.63, irrespective of how you add them up: B (1) C1 (1) + B (1) C2 (1) + B (2) C1 (2) + B (2) C2 (2) = B (1) [C1 (1) + C2 (1)] + B (2) [C1 (2) + C2 (2)]

■

A generalization yields our next result.

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CHAPTER 2: INTEREST RATES

RESULT 2.4 The Sum of a Present Value of Cash Flows Is Equal to the Present Value of the Sum Suppose that there are S securities, which provide s cash flows (s = 1, 2, … , S ), some of which could be zero. These cash flows can occur at times t (t = 1, 2, … ,T), so that the cash flow from security s at time t is Cs (t). Then the sum of the present value of each cash flow is equivalent to (1) adding up the cash flows from all securities at a particular time t, (2) computing the PV of this sum, and (3) adding up these cash flows across time. This can be expressed as T

S

∑∑ t=1 s=1

T

B (t) Cs (t) =

∑ t=1

S

B (t)

[∑ s=1

Cs (t) ]

(2.14)

where B(t) is the price of a zero-coupon bond that pays $1 at time t.

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EXAMPLE 2.4: Finding the Continuously Compounded Rate from a Zero-Coupon Bond ■

Suppose a zero-coupon bond that is worth $0.90 today pays $1 after two years. Assuming that money grows at the continuously compounded rate of r percent per year, we have 0.90e2r = 1 or, e2r = 1/0.90

■

Taking natural logarithms and remembering that log(eX ) = X, 2r = log (1/0.90) or r = (1/2) log (1/0.90) = 0.0527 or 5.27 percent

Replacing numbers with symbols in the preceding expression (2 = T and 0.90 = B = 1/[1 + R]) gives us Result 2.4, which allows us to move from a notional variable (a posted interest rate) to a traded asset price (a zero-coupon bond or a dollar return), and vice versa.

US TREASURY SECURITIES

RESULT 2.5 Computing Continuously Compounded Interest Rates from a Zero-Coupon Bond Price or a Dollar Return The continuously compounded interest rate per year r can be computed from a zero-coupon bond price B (or a dollar return 1 + R) by r = (1/T) log (1/B) = (1/T) log (1 + R)

(2.15)

where T is the time to maturity in years.

Now that you have seen how to price zero-coupon bonds (B), move funds across time (T), find dollar returns (1 + R), and compute continuously compounded interest (r), you may wonder where zero-coupon bond prices come from. This brings us to the Treasury securities market, where debts of the US government trade.

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2.5

US Treasury Securities

Modern finance has made Treasury securities (Treasuries) interesting and exciting by developing a whole range of derivatives that depend on them. Although seemingly ordinary, Treasuries are the bedrock on which the world of finance is built. Following are five reasons why: 1. Five characteristics make Treasuries particularly important: (1) these debt securities have no default risk as they are backed by the full faith and credit of the US government; (2) they trade in a market with some of the smallest bid/ask spreads in the world, which is the transaction cost of buying and selling Treasuries (only 1 or 2 basis points for the most active issues, a basis point being 1/100th of 1 percent); (3) their interest payments are free from state and local taxes; (4) they have low minimum denominations starting at $100; and (5) they offer a spectrum of maturities that range from one day to thirty years. 2. The prices of Treasuries can help us determine borrowing and lending rates for future dates. What is the forward rate for borrowing money for two years starting 10 years from today? As Chapter 21 shows, the answer can be easily determined from Treasury prices. 3. We will study options and futures that are written on Treasuries. Remember the T-bond futures that we mentioned in the first chapter? They are one of the most actively traded futures contracts. The world has become a more volatile place, and

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CHAPTER 2: INTEREST RATES

derivatives based on fixed-income securities (bonds) can be very useful for hedging interest rate risks. 4. Interest rate options pricing models like the Heath–Jarrow–Morton model relax the constant interest rate assumption of the Black–Scholes–Merton model. They often use Treasury rates of different maturities as the necessary inputs. 5. Moreover, in part of a growing trend, financial firms convert many individual loans and debts into a package of securities using a process called securitization and sell them to third-party investors. Your mortgage loans, car loans, and credit card loans may have been financed that way. Loan access to wider markets generates more competitive rates that benefit consumers. The interest rates on these assetbacked securities are determined by adding a basis point spread to comparable maturity Treasury rates.

2.6

US Federal Debt Auction Markets

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When former US president Ronald Reagan was asked for his thoughts on the growing federal deficit, he replied, “The deficit is big enough to take care of itself!” No matter which party held power in Washington, D. C., the last several decades have seen government spending exceed taxes. Years of budget deficits (= Government receipts – Government expenditures) have led to a mountain of federal debt, standing at over $13 trillion at the end of the millennium (Figure 2.3 shows the level of public debt on an annual basis). This is hardly surprising. There is a natural tendency toward overspending in a democracy because elected politicians bring pork-barrel projects to help their constituencies. Whatever the reason, the US national debt and deficit became alarmingly large in the 1990s and took center stage on the political arena. US citizens interpreted continuing deficits as implying higher future taxes. Supply-side tax cuts of Reagan’s era lost their appeal, and federal income tax rates were raised immediately. A robust economy also increased tax receipts. Consequently, the US government started paying off its debts at the turn of the new millennium. But the surpluses soon evaporated, and the budget deficit is back in the red. As circumstances shape destiny, a huge federal debt has forced the Treasury to devise efficient ways of raising funds. Consequently, the United States has developed an extremely sophisticated system of selling Treasuries through sealed bid auctions, a model that has been adopted by many other nations. The buyers in such auctions submit sealed bids. These are promises to buy a fixed number of Treasuries at a fixed price. The Treasury awards the securities to the highest bidders. In reality, these are yield auctions, in which the bidders specify a yield and the Treasury translates this interest rate into a price. Every year, the Treasury finances the public debt through over 250 auctions, each typically selling $10 to $20 billion worth of securities.

US FEDERAL DEBT AUCTION MARKETS

FIGURE 2.3: U. S. Debt Outstanding 20,000 US Federal 18,000 Debt (in billions 16,000 of dollars) 14,000 12,000 10,000 8,000 6,000 4,000 2,000

97 19 99 20 01 20 03 20 05 20 07 20 09 20 11 20 13 20 15

95

19

93

19

91

19

89

19

87

19

85

19

83

19

81

19

79

19

19

19

77

0

Time

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Source: http://www.treasurydirect.gov/govt/reports/pd/histdebt/histdebt.htm.

Primary dealers are large securities firms with whom the New York Fed buys and sells Treasuries to conduct open-market operations that fine-tune the US money supply, examples include Goldman Sachs & Co. LLC, UBS Securities LLC, and Deutsche Bank Securities Inc. These firms also actively bid in Treasury auctions, as do other direct and indirect bidders. These bidders tread cautiously because bidding incorrectly, even several basis points, can make a difference between millions of dollars in profits and losses. So players in the auction submit bids as near to the bidding deadline as possible using all available information. Do all investors in Treasuries take part in competitive bidding? No, because the Treasury allows noncompetitive bids for smaller amounts. These bids are always filled, and they pay the price determined by the competitive bidders. Historically, 10 to 20 percent of a typical Treasury auction has been awarded to noncompetitive bidders. Noncompetitive bids encourage the direct participation of regular folks in the auction process. The quotes from competitive and noncompetitive bids are tallied, securities are allocated to the successful bidders, and the results of the auction are announced in a press conference. The Wall Street Journal and the other news providers carry the announcement.

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2.7

Different Ways of Investing in Treasury Securities

Bidding in auctions is not the only way to purchase Treasury securities. For example, you can buy Treasuries (1) in the when-issued and secondary markets, (2) through the repo and reverse repo markets, or (3) via interest rate derivatives. This section describes each of these alternative ways of buying Treasuries. Barring rare exceptions, these markets tend to be fairly liquid and are usually dominated by big institutional investors (institutions such as commercial banks, investment banks, pension funds, mutual funds, and hedge funds).

The Treasury Auction and Its Associated Markets

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A week or so before an auction, the Treasury announces the size of the offering, the maturities, and the denominations of the auctioned Treasuries. The Treasury permits forward trading of Treasury securities between the announcement and the auction, and the to-be-auctioned issue trades when, as, and if issued. Traders take positions in this when-issued market, and a consensus price emerges. The when-issued market spreads the demand over seven to ten days, which leads to a smooth absorption of the securities by the market. The buyer and seller in the when-issued market fulfill their commitments after the Treasuries become available through the auction. Newly auctioned Treasuries are called on-the-run and have lower spreads than off-the-run securities issued in prior auctions. A very active resale (secondary) market exists for Treasuries, giving investors further chances to invest. The timeline for the three markets for buying Treasuries is shown in Figure 2.4.

FIGURE 2.4: Timeline of Treasury Auction and Associated Markets

Treasury announcement

Auction

When-issued market

Settlement

Post-auction secondary market

DIFFERENT WAYS OF INVESTING IN TREASURY SECURITIES

41

The Repo and the Reverse Repo Market Investment banks operate a security dealership business that makes profits by trading standard securities as well as custom-made products for clients. They do this by highly leveraging both the asset and liability sides of their balance sheets (see Chapter 1 for a typical firm’s balance sheet). For example, believing that buying Treasuries will gain in value and borrowing Treasuries will decline in value, the firm may support $100 billion worth of purchased securities and $95 billion worth of borrowed securities using only $5 billion of capital. Leverages like this are usually accomplished through repo and reverse repo markets, examples of which are given in Extension 2.3. Partially contributing to the immense popularity of repos was the fact that they initially provided the only method of short selling Treasuries and corporate debt securities (see Extension 2.3). This short selling technique became antiquated when interest rate derivatives arrived. Derivatives facilitate the same transaction but far more efficiently. We will explore how this works in the final part of the book. Although this particular economic role of repos is now less important, the repo market still thrives and is an important element of Treasury markets.

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EXTENSION 2.3: A Repurchase Agreement Suppose that Repobank has purchased a large quantity of Treasury securities and it needs an overnight loan to finance the purchase. Repobank expects to sell the Treasuries on the market the next day, so an overnight loan is sufficient. To finance its position, it enters into a repurchase agreement (“does a repo”) with RevRepobank (another fictitious name), who enters into a reverse repo (“does a reverse”) transaction. The next example outlines the mechanics of this transaction in simple terms.

EXT. 2.3 EX. 1: Example of a Repo and a Reverse Repo ■

As per standard industry norms, Repobank takes, say, $10 million from RevRepobank and sells RevRepobank Treasury securities worth a little more. The next day, Repobank repurchases those securities at a slightly higher price—the extra amount determines an annual interest rate known as the repo rate. Though it involves a purchase and a repurchase, a repo is basically a short-term loan that is backed by high-quality collateral (see Ext. 2.3 Fig. 1).

■

If Repobank defaults, then RevRepobank keeps the securities. If RevRepobank fails to deliver the securities instead, then Repobank keeps the cash longer. In that case, the repo is extended by a day, but the terms remain the same—Repobank pays the day after tomorrow the amount it was supposed to pay tomorrow, essentially keeping the funds for an extra day at zero interest. And this is repeated day after day until the cash and the securities are exchanged. You may find that holders of scarce securities are able to borrow funds in the repo market at close to a zero percent rate of interest.

■

Repos provide a way of short selling Treasuries and corporate debt securities. After acquiring the securities, RevRepobank can short sell them to a third party. Later, it can buy these securities in the market, at a lower price if the bet is successful, and return them to Repobank and close out the repo.

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■

Entering into a repo agreement is equivalent to borrowing cash and using the short maturity Treasury security as collateral. The repo rate is the effective borrowing rate for Repobank and the lending rate for RevRepobank.

■

Repos are usually for overnight borrowing and are known as overnight repos. If Repobank wants to borrow funds for another night, then the whole process must be repeated. Alternatively, term repos are set up for a fixed period that lasts longer than overnight, whereas open repos have no fixed maturity and may be terminated by a notice from one of the sides. The market is huge. Major players include heavyweights like the Federal Reserve Bank, state and local governments, commercial and investment banks, mutual funds, and large companies.

EXT. 2.3 FIG. 1: An Overnight Repo and a Reverse Repo Transaction Starting Date (Today)

$10 million Repobank

RevRepobank

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T-securities worth $10 million

Ending Date (Tomorrow)

$10(1 + interest) million Repobank

RevRepobank T-securities worth $10 million

TREASURY BILLS, NOTES, BONDS, AND STRIPS

Interest Rate Derivatives Derivatives whose payoffs depend on Treasury (or closely related) securities are called interest rate derivatives.4 There are derivatives like futures on Treasuries and options on futures on Treasuries that allow investors to place leveraged bets or set up hedges on interest rates. We will discuss interest rate derivatives and their pricing models in Part IV of the book.

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2.8

Treasury Bills, Notes, Bonds, and STRIPS

The Treasury sells three types of marketable debts: (1) coupon bonds that pay interest (coupons) every six months and a principal amount (par or face value) at maturity, (2) zero-coupon bonds that don’t pay interest but pay back the principal at maturity, and (3) floating rate notes that pay interest quarterly. The United States issues bonds with maturities of one year or less in the form of zero-coupon bonds and calls them Treasury bills (or T-bills). Coupon bonds have two names: those with original maturity of two to ten years are called Treasury notes (or T-notes), whereas those with original maturity of more than ten years up to a maximum of thirty years are called Treasury bonds (or T-bonds). These notes and bonds make coupon and principal payments that remain fixed over the security’s life. Treasury notes of two year maturity, which pay interest quarterly based on the 13 week T-bill rate also are issued by the Treasury. There is an exception. In 1997, the Treasury started selling a new class of securities called inflation-indexed bonds or TIPS (Treasury Inflation Protected Securities). These bonds guarantee a fixed real rate of return, which is the nominal rate of return in dollar terms minus the inflation rate as measured by the CPI over their life. This is accomplished by raising the principal of the bond each year by changes in the US CPI. Each year, the coupon payment is computed by multiplying the adjusted (and changing) principal by the real rate of return. The name “coupon” may sound strange to your ears. In days gone by, bonds had coupons attached to them. The bearer of the bond detached the coupon and sent it to the issuer, who mailed back interest payments. Coupons are rare these days. In fact, since 1983, all Treasury securities are kept in book entry form at the Federal Reserve Bank’s computers, and the owner is given a receipt of ownership and receives deposits directly from the Fed. But the term coupon survives. The Treasury does not issue zeros of long maturities. Still, a pure discount bond with a single payment at maturity appeals to many investors. Wall Street firms figured out a way of making money by selling what people demanded. Wall Street firms bought coupon-bearing Treasury securities, put them in a trust to make them safe, and issued claims against the principal and the different coupon payments. This was an arbitrage opportunity: the firms paid less for the original Treasury than what they

4

Closely related to Treasury securities are Eurodollar deposits, which are discussed later in this chapter.

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CHAPTER 2: INTEREST RATES

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collected by selling the artificially created zero-coupon bonds. In February 1985, the Treasury entered this activity by allowing Treasury securities to be STRIPped. How do STRIPS (Separate Trading of Registered Interests and Principal of Securities) work? Although the Treasury does not issue or sell STRIPS directly to investors, it allows financial institutions as well as brokers and dealers of government securities to use the commercial book-entry system to separate a Treasury note or a bond’s cash flows into strips and sell them as individual zero-coupon bonds. Claims to individual cash flows coming out of a Treasury security are synthetically created zerocoupon bonds of different maturities. Moreover, an investor can buy up the individual strips from the market and reconstruct the original T-bond or note. Figure 2.5 shows how thirty-one strips can be created from a newly issued fifteen-year T-bond. A Treasury security can be stripped at any time from its issue date until its maturity date. For tracking purposes, each cash flow due on a certain date is assigned a unique Committee on Uniform Security Identification Procedures (CUSIP) number that depends on its source—consequently, they are categorized as coupon interest (ci), note principal (np), and bond principal (bp). Even though they may come from different Treasury securities, all ci due on a certain date have the same generic CUSIP and may be combined to create a coupon paying note or a bond. By contrast, np or bp is unique for a particular security and hence is not interchangeable. The STRIPS program had been very successful: 1. STRIPS made Treasuries more attractive to investors, leading to greater demand, higher prices, lower yields, and cheaper financing of the national debt. For example, not many people would be interested in holding a thirty-year bond with sixty cash flows. But a newly issued thirty-year bond can be stripped into sixty semiannual coupon payments and a final principal payment. These cash flows can be sold to different investors needing zeros of different maturities. For example, grandparents of a newborn may gift a strip for future education costs. 2. STRIPS help to identify the term structure of interest rates, a graph that plots the interest rate on bonds against the time to maturity. Earlier in this chapter, we talked about the benefits of knowing the risk-free rates of different maturities, an important input to the pricing of interest rate derivatives studied in Part IV of the book. Once we get the price of a Treasury security, we know the cash flows that it generates. Particularly useful are T-bill and STRIPS prices for moving cash flows across time.

TREASURY BILLS, NOTES, BONDS, AND STRIPS

FIGURE 2.5: Creation of STRIPS from a 15-Year Treasury Bond Newly Issued 15-Year T-Bond

Payments Time (years)

C ½

C 1

C 1½

…

…

…

C C and L 14½ 15

C denotes coupons and L principal. Holder gets all the cash flows. STRIP Creation

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Payments Time (years)

C ½

C 1

C 1½

…

…

…

C C and L 14½ 15

C and L sold as separate zero-coupon bonds (STRIPs).

Prices of Treasuries can be found in financial newspapers like the Wall Street Journal and Investor’s Daily. The quotes are collected from the over-the-counter market and are for transactions of $1 million or more. Notes, bonds, and strips are quoted in 32nds, so a quote of 93:08 (for bid) and 93:09 (for ask) for a strip maturing in one year means that $93 and 8/32 is the price the dealer will buy and $93 and 9/32 is the price the dealer will sell. These quotes are based on a par value equal to $100. T-bill prices follow an entirely different convention. They are quoted in terms of a banker’s discount yield. Before introducing this notion, it may be useful to see how the rate of return and the bond-equivalent yield are determined. When the annualized rate of return (see Result 2.1) is applied to Treasury bills, we get its bond-equivalent yield. This is reported as the “ask yield” in the Wall Street Journal, which corresponds to the bond-equivalent yield earned when buying a T-bill from a

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dealer. The bond-equivalent yield for Treasury bill prices is Bond equivalent yield = (

365 (1 − B) T ) B

(2.16)

where B is the bill’s price expressed as a bond that pays $1 at maturity and T is the number of days from settlement to maturity. However, Treasury bills are quoted in terms of a banker’s discount yield, which is typically reported for both ask and bid prices. Ask is the price at which one can readily buy a security from a dealer, and bid is the price at which one can promptly sell. They are analogous to prices at which a car dealer will sell or buy a used car. The banker’s discount yield differs from the bond-equivalent yield in two ways: (1) the denominator has the face value instead of the price paid (note the 1 in the denominator) and (2) the interest rate is annualized on a 360-day basis. The banker’s discount yield is given by Banker’s discount yield = (

360 (1 − B) T ) 1

T or Bill price, B = 1 − (banker’s discount yield) [ ( 360 )]

(2.17) (2.18)

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where 1 is the face value (par value) of the bill, B is bill price, and T is the number of days from settlement (the date ownership is transferred, time 0) to maturity. Example 2.5 applies these formulas to data from the financial press.

EXAMPLE 2.5: T-Bill Price Computation ■

The financial press reports the following: - Days to maturity 24 - Bid 4.71 percent - Ask 4.67 percent - Change 0.01 - Ask yield 4.75 percent

■

Using expression (2.7b) and assuming that the bill’s principal is $1, the ask price for the bill is = 1 − 0.0467 × (24/360) = $0.996887 In other words, $0.996887 is the ask price of a zero-coupon bond that pays $1 in twenty-four days. Similarly, $0.996860 is the bid price of a zero that pays $1 in twenty-four days.

LIBOR VERSUS A LIBOR RATE INDEX

■

How much are we actually earning on our investment? This is given by the bond-equivalent yield (reported as ask yield because it is computed from the ask price). Using expression (2.6) Bond − equivalent yield (for ask yield) = (365/24) × [(1/0.996887) − 1] = 0.0475 This verifies the quote.

The inverse relationship between bond prices and interest rates leads to an inversion of the relation between the ask/offer and bid for interest rates. In most markets, the ask/offer is higher than the bid, but that’s not the case for the ask and the bid when we quote them as a banker’s discount yield. Using the numbers from Example 2.4, this situation is shown in Figure 2.6.

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FIGURE 2.6: Inversion of Ask and Bid When Quoted as Banker’s Discount Yield (Inverted–bid above ask) Bid (as bdy) 4.71 percent

(As in most markets–ask above bid) Ask price $0.996887

Ask (as bdy) 4.67 percent

Bid price $0.996860

The banker’s discount yield is important because it helps price a T-bill. Using the face value in the denominator or 360 days in the computation doesn’t have any economic significance. It’s just the way pricing conventions have developed in this market. Using 360 days for quoting interest rates and using 1 in the denominator (as in Equation 2.2) just make it easier to compute.

2.9

Libor versus a Libor Rate Index

Though US Treasury securities are fundamental to the international bond markets, there are other interest rates that are keenly followed and widely used. In particular, interest rates earned on dollar deposits in non-US banks are such an example. These rates are related to the London Interbank Offered Rate (libor, pronounced lie-bore), which is the interest rate used to determine cash payments on most swap contracts in the swap market. Swaps are discussed in Chapters 7 and 22. To understand how libor gets created, let us peek at the London interbank market. Owing to their business activities, banks and other financial institutions face deficit or surplus fund situations on a regular basis. Banks in the United States manage

47

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48

CHAPTER 2: INTEREST RATES

their cash by borrowing or lending in the interbank federal funds market. Major London banks handle those imbalances by borrowing or lending deposits of different maturities in the London interbank market. The most important of these deposits are Eurodollars, which, as you may recall from Chapter 1, are US dollars held outside the United States in a foreign bank or a subsidiary of a US bank. Owing to the dual benefit of being dollar deposits free from US jurisdiction, Eurodollars have gained huge popularity among a whole range of holders, including central banks, financial institutions, companies, and retail individual investors. Now a bank with surplus funds lends the funds to another bank for a fixed time period at the libor valid for that period. This is a Eurodollar rate (say, 5.02 percent per year for a three-month deposit) in case of Eurodollars. Alternatively, a bank pays the London Interbank Bid Rate (libid, pronounced lie-bid) for the privilege of borrowing money: three-month libid could be 5.00 percent for Eurodollars. These rates may change minute by minute, and they may vary from bank to bank, but competition ensures that they are almost nearly the same at any given point in time. The Intercontinental Exchange (ICE) collects libor quotes from its panel of major banks for deposit maturities ranging from overnight to a year and computes a Libor rate index that is widely reported. For Eurodollar deposits, the ICE collects libor quotes from numerous banks, truncates some of the largest and smallest values, and, averages the rest to compute a libor rate index. It does a similar exercise for deposits denominated in many other currencies, announcing a term structure of Libor rate indexes for each different currency. This libor rate index is the most popular global benchmark for short-term interest rates, and it enters into numerous derivative contracts. An index is used instead of a particular bank’s Eurodollar rate because an index is less prone to manipulation. As the libor rate index has some credit risk, its values are larger than a similar maturity Treasury security, the difference being known as the Treasury/Eurodollar (TED) spread. Instead of Treasuries, many large financial institutions use the libor rate index as a proxy for the risk-free interest rate in derivative pricing models. Like Treasury securities, Eurodollar interest computations assume 360 days in a year. The Eurodollar market is now measured in trillions of dollars. In later chapters, we will study the vast interest rate derivatives market that uses Eurodollars as the underlying commodity.

2.10

Summary

1. An interest rate is the rate of return earned on money borrowed or lent. The annualized rate of return on an investment is given by 365 T ) Selling price + Income − Expenses − Buying price × ( ) Buying price

Rate of return = (

=(

Profit or loss 365 × T ) ( Buying price )

SUMMARY

where T is the time interval, measured by the number of days over which the investment is held, and Income and Expenses denotes any positive or negative cash flows, respectively, from the investment during this period. 2. There are many ways of computing interest. For a certain quoted rate, the realized interest can vary with (1) the method of compounding, (2) the frequency (yearly, monthly, daily, or continuously compounded), (3) the number of days in the year (52 weeks; 360 days, or 365, or 366 for a leap year), and (4) the number of days of the loan (actual or in increments of months). 3. The three basic methods of computing interest are simple, compound, and continuous compounding. In simple interest, an annual rate is used for the loan duration. When computed over several months, a fraction of that annual rate is used. In case of compound interest, interest is computed on both the original principal as well as the accrued interest. Continuous compounding involves paying interest on the principal and accrued interest on a continuous basis.

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4. Interest rates help us find the price of a zero-coupon bond (zeros). These bonds sell at a discount and pay back the face value at maturity. They pay no interest otherwise. Multiply any dollar amount by a zero-coupon bond’s price to get its present value, or divide the amount by it to get its future value. 5. US Treasury securities, whose payments are backed by the taxing power of the US government, are considered default-free. They help us determine the risk-free interest rate. They are sold through a sealed-bid auction, in which all successful bidders pay the same price as the lowest successful bid. One purchases a Treasury security by (1) trading in the auction and its associated markets, (2) entering into a repurchase (repos are short-term debts collateralized by high-quality debt securities) or a reverse repo transaction, or (3) investing in interest rate derivatives. 6. Marketable Treasury securities (Treasuries) are classified as bills, notes, and bonds. Treasury bills are zero-coupon bonds with original maturity of one year or less. Coupon bonds have two names: those with original maturity of two to ten years are called Treasury notes (or T-notes), whereas those with original maturity of more than ten years up to a maximum of thirty years are called Treasury bonds (or T-bonds). 7. Financial institutions and brokers and dealers of government securities can create zero-coupon bonds by segregating cash flows of Treasury notes and bonds and sell them to the public. These artificially created zeros are called STRIPS. 8. In the bond markets, the annualized rate of return is called the bond-equivalent yield. In case of Treasury bills, Bond equivalent yield = (

365 (1 − B) T ) B

where B is the price of a T-bill that pays $1 at maturity and T is the number of days from settlement (the date ownership is transferred, time 0) to maturity. But

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Treasury bills are quoted in terms of a banker’s discount yield, which is given by Banker’s discount yield = (

360 (1 − B) T ) 1

9. Besides Treasuries, major interest rates in the global financial markets include libor and repo rates. Libor and libid are the respective rates for the banks to lend and borrow surplus funds in the London interbank market. The ICE (1) collects libor quotes from major banks for deposits denominated in many different currencies, (2) for each currency it collects quotes with maturities ranging from one day to one year, (3) computes a trimmed mean by dropping some of the highest and lowest rates, and (4) releases these rates to the market, creating a Libor rate index. 10. A libor rate index for Eurodollar deposit constructed by the ICE (which are dollar-denominated deposits held outside the US and free from US banking laws and regulations) has emerged as the most popular global benchmark for shortterm interest rates and enters into numerous derivatives contracts. Many large financial institutions are replacing Treasuries with bbalibor as a proxy for the risk-free interest rate. Many floating interest rates are set at a spread above the relevant Treasury security rate or the Libor rate index.

2.11

Cases

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Breaking the Buck (Harvard Business School Case 310135-PDF-ENG). The case

educates students about how money market funds work and the challenges faced in managing these funds during the financial crisis of 2008–9. Foreign Ownership of US treasury Securities (Darden School of Business Case

UV1366-PDF-ENG, Harvard Business Publishing). The case describes the workings of the US Treasury securities market and explores the implications of significant foreign holdings of US debt. Innovation at the Treasury: Treasury Inflation-Protection Securities (A)

(Harvard Business School Case 204112-HCB-ENG). The case explores the various risks as well as policy issues surrounding the introduction of a new financial security product.

2.12

Questions and Problems

2.1. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a simple interest rate. 2.2. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a compound interest rate with monthly compounding. 2.3. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a continuously compounded interest rate.

QUESTIONS AND PROBLEMS

Time (in years)

Cash Flows (in dollars)

0 (today)

105

1

7

2

9

3

108

2.4. The interest rate is 5 percent per year. Compute the six-month zero-coupon

bond price using a banker’s discount yield (the zero-coupon bond is a US T-bill with 180 days to maturity). 2.5. What is a fixed-income security? The next three questions are based on the

following table, where the interest rate is 4 percent per year, compounded once a year. 2.6. Compute the present value of the preceding cash flows. 2.7. Compute the future value of the preceding cash flows after three years. 2.8. What would be the fair value of the preceding cash flows after two years? 2.9. If the price of a zero-coupon bond maturing in three years is $0.88, what is

the continuous compounded rate of return? 2.10. What are the roles of the primary dealers in the US Treasury market?

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2.11. What is the when-issued market with respect to US Treasuries? What role does

this market play in helping the US Treasury auction securities? 2.12. What is the difference between on-the-run and off-the-run Treasuries? 2.13. What is a repurchase agreement? Explain your answer with a diagram of the

transaction. 2.14. What is a Treasury STRIPS? What benefits do the trading of Treasury STRIPS

provide? 2.15. Explain how a libor rate index is computed by the ICE. 2.16. What is a Eurodollar deposit, and what is a TED spread? 2.17. What is the difference between Treasury bills, notes, and bonds? What are

TIPS, and how do they differ from Treasury bills, notes, and bonds? 2.18. You bought a stock for $40, received a dividend of $1, and sold it for $41 after

five months. What is your annualized arithmetic rate of return? 2.19. Suppose that you are planning to enroll in a master’s degree program two years

in the future. Its cost will be the equivalent of $160,000 to enroll. You expect to have the following funds: ■

From your current job, you can save $5,000 after one year and $7,000 after two years.

51

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52

CHAPTER 2: INTEREST RATES

■

You expect a year-end bonus of $10,000 after one year and $12,000 after two years.

■

Your grandparents have saved money for your education in a tax-favored savings account, which will give you $18,000 after one year.

■

Your parents offer you the choice of taking $50,000 at any time, but you will get that amount deducted from your inheritance. They are risk-averse investors and put money in ultrasafe government bonds that give 2 percent per year.The borrowing and the lending rate at the bank is 4 percent per year, daily compounded. Approximating this by continuous compounding, how much money will you need to borrow when you start your master’s degree education two years from today?

3 Stocks 3.1 Introduction

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3.2 Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets 3.3 Brokers, Dealers, and Traders in Securities Markets 3.4 Automation of Trading 3.5 The Three-Step Process of Transacting ExchangeTraded Securities 3.6 Buying and Selling Stocks Trading at the New York Stock Exchange Over-the-Counter Trading

Alternative Trading Systems: Dark Pools and Electronic Communications Networks

3.7 Dollar Dividends and Dividend Yields 3.8 Short Selling Stocks 3.9 Margin—Security Deposits That Facilitate Trading EXTENSION 3.1 Margin and Stock Trading

3.10 Summary 3.11 Cases 3.12 Questions and Problems

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CHAPTER 3: STOCKS

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3.1

Introduction

Institutions that play similar economic roles can differ widely in appearance. A farmer’s market in a remote African village and a fancy shopping mall in a North American suburb are both markets. MBA degree–holding sellers of customized derivatives products and peddlers of Oriental rugs are both dealers. The man at the airport who helps you find a cheap hotel near an exotic beach and the realtor who arranges the purchase of a vacation home are both brokers. As these examples illustrate, seemingly diverse economic phenomena often have commonalities. This leads to this chapter’s theme—“unity in diversity”; here we try to understand the features common to different securities markets and their traders.1 We introduce primary and secondary markets, exchanges and over-the-counter markets, brokers and dealers, and the bid and ask prices that dealers post. We also discuss market microstructure, a subfield of finance that studies how market organization and traders’ incentives affect bid/ask spreads. We classify traders into different categories on the basis of their trading strategies. We describe how automation is transforming trading and the three-step process of execution, clearing, and settlement that is common when transacting exchange-traded securities. Next, we study the effect of dividends on stock prices and portfolios, paving the way for managing a portfolio that replicates a stock index and pricing derivatives written on it. Finally, we illustrate the process of short selling stock and margins. The process of trading securities, the players who trade, and the facilitators who make trading possible are also present in the bond markets discussed in Chapter 2. However, we discuss them with respect to stock markets first because these are the easiest markets to understand—trading of different derivatives can also be seen as extensions of this process.

3.2

Primary and Secondary Markets, Exchanges, and Over-the-Counter Markets

As a small company grows, it needs more and more cash to finance its operations. After initial financing from founders and friends, and perhaps private equity from venture capitalists, the company becomes ready to tap the capital markets. A privately held company sells shares to the public for the first time in an initial public offering (IPO), which is the primary market for stocks. To issue stocks, you hire an investment banker who manages the process. Through cumbersome paperwork, publishing information brochures, and getting approval from government agencies, the investment banker burns a lot of your future proceeds in the process. Soon after their birth in primary markets, stocks move on to secondary markets, where they change ownership through secondhand transactions. Exchanges and the over-the-counter (OTC) market are the two basic types of secondary markets. 1

The phrase “Unity in Diversity” is the official motto of the European Union (“United in Diversity”), Ghana, Indonesia, South Africa, and some other nations; it has also been used to describe India.

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PRIMARY AND SECONDARY MARKETS, EXCHANGES, AND OVER-THE-COUNTER MARKETS

Traditionally, the main characteristic of an exchange was a central physical location where buyers and sellers gathered to trade standardized securities under a set of rules and regulations, the New York Stock Exchange (NYSE) being a prime example. However, the information revolution of the 1990s changed that forever— most exchanges in the new millennium, especially those outside of the US, do not have floor trading; instead, they have a centralized computer network that does trade matching. By contrast, any trade away from an exchange is called an OTC transaction.2 The expression over-the-counter goes back to the early days of the US, when banks primarily acted as dealers for stocks and bonds, which were sold at counters in their offices. There are organized markets for OTC transactions, where commercial and investment banks, institutional investors, brokers, and dealers participate. Such OTC markets (also called interbank markets) have no central location. Telephone, telex, and computer networks connect geographically dispersed traders and make trades possible in OTC markets. Exchanges used to handle the bulk of stock trading in the US, but in recent times, organized OTC markets have taken on increased importance. Exchanges operate under the basic assumption that if all trades take place in a central location, this provides a level playing field that leads to fair price formation. US stock exchanges are regulated by the Securities and Exchange Commission (SEC), a federal government agency whose job is to protect investors, punish violators, and prevent fraud. Before the SEC’s formation in 1934, the stock exchanges were considered a hotbed of shady activity that routinely hurt the ordinary investor. Mark Twain famously said in Pudd’nhead Wilson’s tale (1894), “October. This is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August and February.” Interestingly, many folks no longer consider stock exchanges as bad or risky; rather, they are now viewed as venues where respectable investors pursue wealth by taking acceptable levels of risk. The derivatives exchanges, riding a wave of mysticism and intrigue, have easily snatched the disrepute. All exchange-traded securities, including derivatives, must satisfy government requirements before they can be sold to the public or traded on an exchange. For example, a federal government regulatory agency called the Commodity Futures Trading Commission (CFTC) approves every kind of futures contract before it sees daylight. Moreover, for a security to trade on an exchange, the issuing company must satisfy some listing requirements in terms of the company’s assets, annual earnings, shareholder interests, and audit requirements, among other factors. The NYSE has the steepest requirements among the American exchanges, and the US listing requirements are generally stricter than those in Asia or Europe. Exchanges tend to have strict rules and regulations, codes of conduct for members, and self-governance procedures. By lowering the likelihood of market manipulation and fraud, exchanges make traders comfortable, which increases trade volume (business). No exchange likes to have a government regulator at close heels, and a good self-governance program keeps them away. Moreover, self-regulation can stave 2

Most OTC transactions involve a dealer who stands ready to buy and sell securities from an inventory that she maintains. The next section discusses dealers in greater detail.

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off legal actions by angry customers who may otherwise feel cheated. Self-regulation is costly, but like a vaccination, its benefits outweigh the costs. Most exchanges also have a system of mediating disputes (arbitration) that handles problems early in the process and lowers the chances of lawsuits. Although small offenses may avoid detection, the big ones tend to get punished. Regulators and exchanges regularly report names of guilty individuals and their punishments. The press disseminates this information to the general public. OTC contracts require no regulatory approval. Organized OTC markets, being an electronically connected network of spatially separated traders, have fewer restrictions than an exchange. They offer investors greater investment choices but little transaction safety. For many OTC markets, when the going gets tough, the tough may skip town—there is risk of a counterparty failing to honor his side of the contract. In OTC markets, as in life, you need to remember the adage “know your customer.”

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3.3

Brokers, Dealers, and Traders in Securities Markets

Want to buy a house or get insurance protection? Call a broker. Need a new car or planning to purchase a fancy electronic gadget? See a dealer. Because people trade at different times, and they desire to see a collection of goods to make their pick, dealers are useful. Dealers intermediate across time by maintaining an inventory of goods. Buyers can buy from and sellers can sell to a dealer. The dealer sells a commodity at the offer or the ask price and buys the commodity at the bid price. Ask and bid prices were defined in Chapter 1. (Write a [for ask] above and b [for bid] below; Ask is the Buying price for the Customer [abc].) For example, a used car dealer sets two prices: the retail (or the ask) price at which he sells cars and a wholesale (or the bid) price at which he buys cars. The dealer earns a living from the difference between these two prices, which is known as the bid/ask (bid/offer) spread. In finance, spread has four different uses: (1) the gap between bid and ask prices of a stock or other security, (2) the simultaneous purchase and sale of separate futures or options contracts for the same commodity for delivery in different months, (3) the difference between the price at which an underwriter buys an issue from a firm and the price at which the underwriter sells it to the public, and (4) the price an issuer pays above a benchmark fixed-income yield to borrow money. These spread definitions (except for the third definition, which is more relevant in a corporate finance course) will be used throughout the book. Paucity of information concerning trading opportunities is an impediment to transactions. Brokers are intermediaries who help to overcome this hurdle and facilitate transactions. They match buyers and sellers and earn commissions for this service. Brokers have no price risk because they carry no inventory and do not trade on their own accounts. Dealers face price risks because they hold inventories. In both exchanges and OTC markets, brokers and dealers play significant roles. They earn a living by maintaining smoothly functioning, orderly markets. Many exchanges designate specialized dealers as market makers. They make markets

BROKERS, DEALERS, AND TRADERS IN SECURITIES MARKETS

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by posting ask and bid prices and stand ready to trade at those prices throughout the trading day—and enjoy enhanced trading privileges for their services. In OTC markets, some dealers take up market-making functions. Brokerage and dealership are risky business—charging too much will drive away customers, whereas charging too little will wipe the business out. Risks originate from many sources: ■

Set security/margin deposit levels. Higher margins make brokers safer, but they are costlier to customers.

■

Inventory to maintain. A larger inventory provides more to sell, but it’s costlier to maintain and makes the dealer more vulnerable to price declines.

■

Bid and ask prices to post. A narrower spread means more transactions but fewer profits per trade, while a widening of the spread has the opposite effect.

■

The amount of securities to bid or offer at those prices. A higher amount at any price exposes the dealer to a greater risk of sharp price movements.

Brokers and the dealers understand these trade-offs and finely balance risks and expected returns to survive in ruthlessly competitive markets. The setting of bid and ask prices is the most important decision that a dealer faces. There are many ways of characterizing traders in security markets. One common practice is to distinguish them as individual investors or institutional traders. For our purpose, it is more useful to categorize them in terms of their trading strategies either as arbitrageurs, hedgers, or speculators. Arbitrageurs seek price discrepancies among securities and attempt to extract riskless arbitrage profits. Hedgers try to reduce risk by trading securities and are often cited as the chief reason for the existence of derivatives markets. Speculators take calculated risks in their pursuit of profits; they may be classified as scalpers, day traders, or position traders on the basis of how long they hold their trades: 1. Scalpers trade many times a day with the hope of picking up small profits from each transaction. They are physically present on an exchange’s trading floor and pay very low trading costs. Scalpers’ holding periods are often measured in minutes and seconds. Market makers are scalpers. Scalpers make markets more liquid by standing ready to buy and sell at posted prices. Liquidity is a desirable feature of securities markets. It’s the ease with which an asset can be converted to cash, and vice versa. 2. Day traders try to profit from price movements within the day. They open their trading positions in the morning and close them out before going home at night. 3. Position traders (also called trend followers) maintain speculative trading positions for longer periods of time. Position traders often try to identify and capture the abnormal price differences between two assets (a spread), with the hope of reaping a profit when the spread is restored to its historical values. These distinctions aren’t ironclad, and traders sometimes flit from one role to another.

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3.4

Automation of Trading

Toward the turn of the century, rapid advancements in IT gave us revolutionary products and services, for example, cell phones, the Internet, and personal computers. Exchanges, brokers, and dealers exploiting these technologies have lowered trading costs and in many cases eliminated human labor from the trading process. IT has also revolutionized trading: innovations like electronic exchanges, online brokerage accounts, and network linkages among exchanges are now commonplace.

3.5

The Three-Step Process of Transacting Exchange- Traded Securities

Security trading on an organized exchange generally involves three steps: execution, clearing, and settlement. OTC transactions follow a similar approach, except that there is no clearing done by a third party (Figure 3.1 shows these three operations for a typical stock trade):

FIGURE 3.1: Execution, Clearing, and Settlement in the Stock Market A) Execution

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Buyer

Seller

Broker

Agree on trade

Broker

(Alternately, a dealer may replace a trader and her broker) B) Clearing (after market closing but before market opening on next trading day) Brokerage firm representatives meet with clearinghouse officials to “clear” (recognize and record) a trade. C) Settlement (one or more business days after execution) Buyer pays seller and gets ownership of the securities.

BUYING AND SELLING STOCKS

■

Execution. A trade is executed when the buyer (or her representative broker) and the seller (or his rep) “meet” on an exchange, physical or electronic; agree on price and quantity; and commit to trade.

■

Clearing. Before the market reopens, an executed trade must clear through the exchange’s clearinghouse. Some brokers and dealers become clearing members, who, besides clearing their own trades, clear trades for their clients and for nonclearing brokers and dealers. The clearinghouse clears a trade by matching the buy and sell orders, recognizing and recording the trade. The clearinghouse in a derivatives exchange performs the additional role of guaranteeing contract performance by becoming a seller to each buyer and a buyer to every seller.

■

Settlement. Finally, a trade ends with a cash settlement when the buyer pays for and gets the securities from the seller. In the old days, this involved the exchange of an ownership certificate for cash or check. Nowadays, it is usually done through transfer of electronic funds for ownership rights between brokerage accounts.

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IT advances have drastically reduced the time lag between these three steps. Trades can be executed in the blink of an eye and even settled on a real-time basis. While the NYSE and NASDAQ remain dominant venues for stock trading, there are more execution choices than ever before. Your broker has a “duty to seek the best execution that is reasonably available for its customers’ orders” (see www.sec.gov/investor/pubs/tradexec.htm). For example, a broker may send your order: ■

to trade an exchange-listed stock. to a national exchange (like the NYSE), to a smaller regional exchange, or to a firm called a third market maker, which stands ready to trade the stock at publicly announced prices

■

to trade an OTC stock to a market maker in the OTC markets

■

to an electronic communications network (ECN), which is an electronic system that automatically matches buy and sell orders at specified prices

■

to another division of his firm to internalize the order by filling it from the firm’s own inventory

You can also instruct the broker to route your order to a specific exchange, a particular dealer, or a predetermined ECN.

3.6

Buying and Selling Stocks

Stocks have been trading in Europe for centuries. The oldest stock exchange opened in Amsterdam in 1602 with printed shares of the United East India Company of the Netherlands. The London Stock Exchange traces its origins back to 1698, when a list of stock and commodity prices called “The Course of the Exchange and other things” was issued, and stock dealers, expelled from the Royal Exchange for rowdiness, started to operate in the streets and coffeehouses nearby (see the London Stock Exchange’s website at www. londonstockexchange. com). The nineteenth century belonged to

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England—Britannia ruled the waves, the sun never set on the British Empire, and London was the commercial capital of the world. After World War I, the hub of commercial and trading activity shifted from London to New York and to the NYSE.

Trading at the New York Stock Exchange The NYSE began in 1792, when 24 prominent stockbrokers and merchants assembled under a buttonwood tree on Wall Street in Lower Manhattan and signed an agreement to trade charging a uniform commission rate. Eventually, the Big Board (a popular name for the NYSE) became the world’s best known and one of the largest stock markets. The exchange heavily invests in technology and has pioneered many innovations such as the stock ticker, an automated quotation service, electronic ticker display boards, and electronic order display books. Example 3.1 discusses a hypothetical trade on a typical exchange.

EXAMPLE 3.1: Exchange Trading of Stocks

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Trade Orders ■

Suppose Ms. Longina Long wants to buy a round lot (one hundred shares) of YBM, which is the abbreviation for Your Beloved Machines Inc., a fictitious company that we use throughout this book. Assume that like many major US companies, YBM trades on a stock exchange. Ms. Long has opened a brokerage account and placed enough funds into it for trading. She places an order by calling up her broker or submitting the order through an Internet account.

■

The broker’s representative records and time stamps her order; web orders automatically record this information. Smaller orders are usually executed quickly. Block trades that involve ten thousand shares or more are often negotiated away from the trading floor, with or without the help of brokers.

■

Various types of orders can be submitted. A limit order must be filled at the stated or a better price or not traded at all. By contrast, a market order must be immediately transacted at the best available price. A liquid stock is actively traded and has many limit buy and sell orders around the market price. As such, reasonably sized stock positions can easily be converted into cash with a minimum loss of value.

Trade Execution ■

Suppose Long submits a market order and a counterparty is sought to take the other side of the trade. This could be another investor or a professional. Long sells 100 YBM stocks at the ask price of $100.10 per share, and the trade gets executed. Ms. Long pays $10,010 for shares plus brokerage commissions. A discount broker may charge her as little as $5, while a full-service broker will cost more.

Clearing and Settlement ■

After execution, most exchanges transmit trade records via computer to a little known but powerful organization called the National Securities Clearing Corporation (NSCC) for clearance and settlement. They are sent as locked-in transactions, which already have trade details from buyers and sellers prematched by a computer. The NSCC also “nets trades and payments among its participants,

BUYING AND SELLING STOCKS

reducing the value of securities and payments that need to be exchanged by an average of 98% each day.”3 It generally clears and settles trades on a two-business-day cycle (T + 2 basis). 3

www.dtcc.com/about/subs/nscc.php.

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Over-the-Counter Trading Over-the-counter markets, along with regional exchanges, have always provided the major venue for trading small stocks and other securities that could not be listed on the Big Board. Since the 1970s, NASDAQ has been considered the major OTC market in the US. Following the Penny Stock Reform Act of 1990, the OTC Bulletin Board (OTCBB) began operations as “a regulated quotation service that displays real-time quotes, last-sale prices, and volume information in over-the-counter (OTC) equity securities.”4 Subsequently, the OTC market in the US has shifted to the OTCBB and the Pink Quote OTCBB, which has expanded over the years, bringing greater transparency to the OTC equities market. Only market makers can quote securities in the OTCBB, and FINRA rules prevent them from collecting any fees for this service. OTCBB displays market data through vendor terminals and websites. The Pink Quote (formerly known as Pink Sheets owing to the color of the paper on which the quotes used to be printed) is an electronic system that displays quotes from broker-dealers for some of the riskiest stocks trading in the OTC markets. The SEC website cautions potential traders, “with the exception of a few foreign issuers, the companies quoted in Pink Quote tend to be closely held, extremely small and/or thinly traded. Most do not meet the minimum listing requirements for trading on a national securities exchange. . . . Many of these companies do not file periodic reports or audited financial statements with the SEC, making it very difficult for investors to find reliable, unbiased information about those companies.”5 Caveat emptor, or “buyer beware,” is the guiding principle in this market.

Alternative Trading Systems: Dark Pools and Electronic Communications Networks The technology revolution has introduced securities trading in venues that are opaque. These new trading venues are called alternative trading systems (ATS). ATS such as dark pools and electronic communications networks (ECNs) are SECapproved non-exchange-based markets for trading securities. Such nonexchange trading venues are a result of the SEC’s 1998 Regulation ATS (Regulation of Exchanges and Alternative Trading Systems). Dark pools (or dark pools of liquidity) have several features. First, they are secretive trading networks that do not send an order directly to an exchange or display 4

www.otcbb.com/aboutOTCBB/overview.stm.

5

www.sec.gov/answers/pink.htm.

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it in a limit order book. Second, an interested trader either negotiates with a potential counterparty or gets matched with one by the dark pool. Third, the market is a nearexclusive domain for institutional players with large orders. National exchanges like NYSE and the NASDAQ Stock Market also route some of their orders to dark pools. ECNs are alternate trading systems that are registered with the SEC as brokerdealers. An ECN’s participants are subscribers, which include institutional investors, broker-dealers, and market makers. Individual traders can indirectly participate by opening an account and submitting trades through a broker-dealer subscriber. ECNs primarily trade stocks and currencies. Trades are usually submitted as limit orders, which get executed when the ECN matches buy and sell orders according to some protocol. If no matching order is found, an ECN may send it to another market center for execution. Unlike a dark pool, the orders are publicly displayed to all subscribers. Founded in 1969 to provide electronic trading for institutions, Institutional Networks (Instinet) has been a pioneer in electronic trading and it became the first ECN in 1997. The NASDAQ (founded in 1971) may be considered an early ECN. ECNs were formally recognized and approved in 1998 when the SEC introduced Regulation ATS. By slashing trading fees and providing better execution, ECNs have enjoyed mind-boggling growth.

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3.7

Dollar Dividends and Dividend Yields

“All men are ready to invest their money, but most expect dividends,” observed T. S. Eliot in Choruses from the Rock in 1934. As Eliot observed, dividend payments in the form of cash (cash dividend) or additional shares (stock dividend) are a timehonored way for companies to reward their shareholders. For dividend payments, the ex-dividend date is an important day. If you buy the stock before the exdividend date, you get the share and the dividend (buying stock cum-dividend); however, if you buy it on or after this date, then you get the stock without the dividend (buying stock ex-dividend). Let us analyze the behavior of the stock price on the ex-dividend date. Suppose the cum-dividend stock price is $100 and the company pays a $2 dividend. Then, we claim that the ex-dividend stock price should be $98. Why? If it were not, profithungry traders would take advantage of any deviation, and as profits and losses occur, this trading activity eventually eliminates the discrepancy from the market price. To see how this works, suppose the ex-dividend stock price is $99 instead. Then, clever traders will buy this stock just before it goes ex-dividend. Their $100 investment will immediately become $101 (ex-dividend stock price $99 plus dividend worth $2), which they can sell and make $1 in instant profit. Conversely, if the stock price falls by more than $2, say, it falls to $97 after the stock goes ex-dividend, then the traders can “go short” (borrow and sell the stock) and buy back the stock after it goes ex-dividend, locking in $1 as instant profits. In this way, $98 is the only exdividend stock price consistent with no riskless profit opportunities. We formalize this observation as a result. (see Result 3.1).

DOLLAR DIVIDENDS AND DIVIDEND YIELDS

RESULT 3.1 The Relation between a Stock’s Price Just Before and After It Goes Ex-Dividend In a market free of riskless profit opportunities,a SCDIV = SXDIV + div

(3.1)

where SCDIV is stock price cum-dividend, SXDIV is stock price ex-dividend, and div is the amount of the dividend. a

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Later, we will call this an arbitrage opportunity. Formally, this result also depends on the stock price process having a continuous sample path (see Heath and Jarrow 1988). Of course, this result also depends on the market satisfying the standard frictionless and competitive market assumptions that are introduced in later chapters.

Dividends are paid in dollars. However, to compare dollar dividends across stocks with different valuations, one computes dividend yields. A dividend yield is the dividend expressed as a fraction of stock price. In this example, the dividend yield is (2/98) = 0.0204 or 2.04 percent. Dividing by the stock price normalizes the dollar dividend and makes a comparison possible across stocks on an apples-to-apples basis. The dividend yield is also useful when dealing with stock indexes like the Standard and Poor’s 500 Index (S&P 500), which is a weighted-price average of five hundred major US stocks (indexes are discussed in more detail in Chapter 6). Accounting for hundreds of dividends (near two thousand dividends in the case of the S&P 500) and adjusting the index’s value accordingly are daunting tasks. A simple technique for making this adjustment using the dividend yield involves the following steps: (1) add up all the dividends paid over the year and express this sum as a percentage of the index value generating a dividend yield, and call it 𝛿; (2) assume that 𝛿 is paid out over the year on a continuous basis, proportional to the level of the index; and (3) assume that the dividend gets continuously reinvested in the index so that the index grows in terms of the number of units. An initial investment of one unit in this index then becomes eᄕ units after one year and eᄕT units after T years. We illustrate this computation with Example 3.2.

EXAMPLE 3.2: Dividend Yields ■

Suppose that a fictitious stock index named “INDY Index” stands at 1,000 on January 1 and has a dividend yield of 𝛿 = 0.06 or 6 percent per year. Assume that dividends are paid m = 3 times a year, where each payment is 𝛿/m = 0.02 times INDY’s price at the dividend payment date. Let us buy one unit of the index “INDY Spot.”

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■

Four months later, INDY stands at 1,100. The dividend payment is 0.02 times this amount, which equals 22. As we reinvest this dividend, the amount of the dividend is irrelevant—all we need to know is that we have 1 × 1.02 = 1.02 units of the index from now onward.

■

Four more months later, we will get another 2.00 percent of the index level as a dividend. Because we were previously holding 1.02 units of INDY Spot, we now have 1.022 = 1.0404 units of INDY Spot. Similarly, at the end of the year, we have 1.023 = 1.061208 units of the index.

■

Next, we modify the payout frequency and assume that the dividend is paid continuously and plowed back into the portfolio after each payment. Here an investment of one unit of INDY Spot will be worth eᄕT units after T years (see Appendix A). Eight months later, we have eᄕT = e[0.06×(2/3)] = 1.040811 units of the index. This property is restated as Result 3.2.

RESULT 3.2

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Valuing an Asset Paying a Continuously Compounded Dividend Yield Consider an asset that pays a continuously compounded dividend yield of 𝛿 per year, which is reinvested back in the asset. Then, a unit investment in the asset today grows to eᄕT units after T years.

Note that Result 3.2 gives a quantity adjustment to the stock price to incorporate a dividend payment, while Result 3.1 is a price adjustment. These two modifications will be used later in the book when considering stocks paying dividends.

3.8

Short Selling Stocks

There are two ways of selling shares: selling a stock from one’s existing portfolio and selling short a share that one does not own. The first transaction is as straightforward as stock buying, whereas the second involves a more complex procedure. A short seller borrows shares from an existing stockholder and then sells them. He has the obligation to buy back the shares in the future and return them to the lender. Moreover, he owes the lender any dividends that the company pays. Interestingly, the act of short selling artificially creates more long shares than were originally issued in the primary market. Later in the book, you will see that this can become a critical factor in market-manipulation trading schemes.

SHORT SELLING STOCKS

A stock buyer is bullish because she expects that the stock will go up in value. A stock seller is bearish because he expects the stock to fall in value. A short seller is much more aggressive in his bearish stance, unlike a seller, who simply gets rid of a stock. A short seller bets on his negative view by undertaking a fairly risky trade, which could be quite dangerous if the bet goes wrong. Chapter 10 discusses an example in which a manipulator corners the market and squeezes the shorts. To see how short selling works, let’s walk through a short sell example (see Example 3.3).

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EXAMPLE 3.3: Short Selling ■

“Typically when you sell short, your brokerage firm loans you the stock. The stock you borrow comes from either the firm’s own inventory, the margin account of other brokerage firm clients, or another lender,” informs an article on the SEC’s website.7

■

Suppose Mr. Oldham owns one hundred shares of BUG (Boring Unreliable Gadgets, a fictitious company name), which are held in his account at Mr. Brokerman’s firm. To make them available for short selling, the shares are held in “street name.” Oldham doesn’t gain anything from this—he is just helping Mr. Brokerman earn some commissions. For simplicity, let’s assume that Mr. Brokerman is the only broker in town.

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Brokerman borrows one hundred shares from Oldham, lends them to Shorty Stock, and helps Shorty sell them short to Mr. Newman (see Figure 3.2). In the future, Shorty has to buy one hundred BUG shares, no matter what the price is, and give them back to Oldham. He owes any dividends paid over this period to Brokerman, who credits them to Oldham’s account.

FIGURE 3.2: Short Selling Example

Brokerman Borrows

Oldham Owns 100 BUG shares in street name; loses voting rights

Pays dividends

Newman Owns 100 BUG shares Dividends from BUG Short-sells

Shorty Stock Shorty sold 100 BUG shares short

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The market is now long two hundred and short one hundred BUG stocks. Still, from BUG’s perspective, there are only one hundred outstanding shares owned by Newman. The other one hundred shares held long by Oldham and the one hundred shares short sold by Shorty Stock are artificial creations, for they cancel one another. Newman gets dividends from BUG and has the voting rights.

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Strangely, Oldham’s original shares became artificial shares during the short selling process. They are Shorty Stock’s babies, which he has to look after. Thus Shorty has to match any cash dividends, stock dividends, and so on, that Oldham would have rightfully received.

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Dividends lower a stock’s value and benefit Shorty Stock because he can now buy them more cheaply. A dollar dividend on a $50 stock will lower its value to $49, which will help Shorty earn a dollar if he now buys back the stock. Paying Oldham this dollar will keep everyone happy by making things fair.

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The only thing that Oldham really loses is voting rights because BUG will not recognize an extra owner. He doesn’t mind that, unless he is planning to thrust major changes on BUG through proxy fights.

7

“Division of Market Regulation: Key Points about Regulation SHO” describes the short selling process for retail customers (see www.sec.gov).

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3.9

Margin—Security Deposits That Facilitate Trading

The word margin has many meanings. In economics, the terms margin and marginal are often used to denote what happens as a result of a small or unit change. In accounting, the term profit margin (of a business) refers to the amount by which revenues exceed costs. In financial markets, margin means “a sum deposited by a speculator with a broker to cover the risk of loss on a transaction on account; now esp. in to buy (also trade, etc.) on margin” (according to the Oxford English Dictionary Online). Margins and collaterals are fundamental to any security market transaction involving explicit or implied borrowing. Let’s consider some examples: ■

Buying stocks on borrowed funds. You decide to leverage (usually called gearing in Europe) your portfolio to increase your potential returns. Suppose you invest $10,000 of your own funds and take a loan of $8,000 from your broker to buy securities worth $18,000. You must open a margin account (also known as a cash and margin account) with your broker. Your investment of $10,000 is the margin or collateral, which acts like a security deposit. Now, a 10 percent return on your leveraged portfolio will be $1,800 earned on your investment, an 18 percent return. But leverage cuts both ways because losses also get magnified. Your actual return is lower because the broker charges you interest on the daily balance of your loan until you pay him back.

MARGIN—SECURITY DEPOSITS THAT FACILITATE TRADING

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67

Short selling stocks. A short seller must also open a margin account. The broker requires you to keep the proceeds from the short selling plus additional funds as margin in this account. Remember that to short, you borrowed the stock first.

Before initiating a trade that requires margin, you must put up the initial margin. This could be in the form of cash or high-quality, low-risk securities. For the broker’s protection, there is a maintenance margin requirement: the minimum amount that you must maintain in the account to keep it open. If your position loses value and the margin account balance declines to this threshold level, the broker will issue a margin call that requires you to promptly provide enough cash (maintenance margin is always in the form of cash) to restore your account balance to the maintenance margin level. If you fail to do this, the broker liquidates your portfolio. Extension 3.1 shows how margin account adjustments work.

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EXTENSION 3.1: Margin and Stock Trading As a result of the great stock market crash of 1929, the US government would like to control the level of stock speculation and has entrusted the Federal Reserve Board with the job of setting minimum margin requirements for securities trading. The exchanges and brokerage firms may set them at even higher (but not lower) levels. However, the real role of margin in financial transactions is to minimize counterparty credit risk. Currently the Fed sets the initial margin requirement at 50 percent and the maintenance margin at 25 percent for stocks (and convertible bonds). This means that when initiating a margin trade, you must come up with at least 50 percent of your own funds and finance the rest with a loan from your broker. Your purchased securities will appear with a credit sign and the loan with a debit mark in your brokerage account, and the broker will charge interest on it. As it is a secured loan (a low-risk loan backed by collateral; the market value of the stock is much more than the amount borrowed), your broker will probably charge the call loan rate (or broker’s call) and add a slight premium. Adding a spread to the benchmark libor index rate usually determines the broker loan rate. It is regularly published (along with other interest rates) in the Wall Street Journal and in the business sections of many other newspapers. The following formula is used for margin computations.

Formula 1: Margin Computation for Stock Trading The margin (or security deposit) computation in connection with buying stocks on margin (by taking a loan from a broker) or short selling is given by the formula Margin = [(Market value of assets –Loan) /Market value of assets] If maintenance margin is 25 percent, then your equity must be at least 25 percent of the portfolio value. If your equity has dropped below this level, you must respond to a margin call, deposit additional funds, and bring the equity back to the 25 percent level. Your firm may set its own margin requirements (often called “house requirements”) at a higher level, say, 40 percent, for maintenance margin. Margin adjustments can get very complicated (see Sharpe, Alexander, and Bailey 1999). The following example illustrates the computations involved (see Ext. 3.1 Ex. 1).

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EXT. 3.1 EX. 1: Margin Account Adjustments for Stock Trading

Buying Stocks on Margin ■

Suppose you buy two hundred YBM stocks at $100 each. The purchase price is $20,000, and you decide to borrow 50 percent of this amount, $10,000, from your broker. Your investment is $10,000. By our formula, initial margin is given by (20, 000 − 10, 000) /20, 000 = 0.5 or 50 percent

■

Suppose YBM goes up to $110. Then - the market value of your assets is 200 × 110 = $22,000 - the loan amount is still $10,000 - the investment in your account is (Market value of assets – Loan) = 22,000 – 10,000 = $12,000 - the margin in your account = 12,000/22,000 = 0.5455

■

Suppose YBM goes down to $60. Then - the market value of the portfolio is 200 × 60 = $12,000, and your investment is 12,000 – 10,000 = $2,000 - the margin = 2,000/12,000 = 0.1667 - you will receive a margin call to come up with more cash, an extra $1,000, so that your margin meets the 25 percent requirement

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Margin Account for a Short Seller ■

Suppose you decide to sell YBM short instead and hope to use the $20,000 in proceeds to buy a new red car. However, the broker not only refuses to let you touch any of this money but also requires you to deposit an additional 50 percent, or $10,000, into your margin account. Isn’t this outrageous?

■

If you think carefully, the broker’s actions are entirely justified. Consider what would happen if he keeps just $20,000 and the stock price goes up by just 1 cent. Your account now has a negative value of – 200 × 100.01 + 20,000 = –$2. The 50 percent initial margin requirement provides the cushion he needs. Whereas a stock buyer may or may not open a margin account, a short seller is always required to do so.

■

In contrast to a regular stock purchase, a short sale involves an initial sell and a subsequent buy. Consequently, any increase in the stock price will increase the liability and eat up margin in your account.

3.10

Summary

1. This chapter has two themes: a study of securities trading in general and stock trading in particular. Institutional details matter in finance. Stock trading is the easiest type to understand, and the trading of different derivatives can be seen as extensions to this process. 2. Stocks first trade in primary markets and then in secondary markets. 3. Exchanges were characterized by central physical locations where buyers and sellers gathered to trade. Nowadays, many exchanges are completely electronic and have a central computer executing the trades.

SUMMARY

4. Any trade away from an exchange is called an over-the-counter transaction. There are organized OTC markets (interbank markets), which are diffused networks of buyers and sellers, brought together by telecommunication connections. 5. Brokers match buyers and sellers and earn commissions for this service. Dealers trade on their own account and survive by posting a bid/ask spread. 6. Traders may be classified into three categories: hedgers who use securities for risk reduction, speculators who accept risky trades with profit expectations, and arbitrageurs who try to find price discrepancies and extract riskless arbitrage profits with strategies like buying low and selling high. Speculators can be further distinguished as scalpers, day traders, and position traders on the basis of how long they hold their trades. 7. The IT revolution has given us online brokerage accounts, electronic exchanges, and network links among exchanges and has equipped individual investors with unprecedented resources. Electronic exchanges minimize human involvement in the trading process and have eliminated floor trading altogether. 8. Security trading in an organized exchange generally involves three fundamental steps: execution, clearing, and settlement. There are more execution choices today than ever before. A trade may be sent to a national exchange, to a smaller regional exchange, to a third market maker, to a market maker in the OTC market, to an alternative trading system such as an electronic communications network, or to another division of the firm.

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9. The NYSE is a prototypical securities market. It merged with an electronic exchange and demutualized and restructured itself as a for-profit company. 10. Dividend payments are a time-honored way of rewarding stock investors. When the stock goes ex-dividend, the cum-dividend stock price equals the ex-dividend stock price plus the dividend. 11. The concept of continuously compounded interest can be used to study an investment in a portfolio that mimics the cash flows from a stock index. Consider one unit investment in an asset that pays dividends according to a continuously compounded yield 𝛿 per year, which is reinvested back in the asset (assume dividend 𝛿/m is paid and compounded m times per year, where m becomes infinitely large). Then, after T years, the portfolio will contain eᄕT units of the asset. 12. A short seller borrows shares held in street name and sells them short. The original owner loses voting rights, and the short seller compensates the owner for dividends. 13. Although the concept of margin is similar for trading different securities, the transactions vary. Traders buying stocks open a margin account only if they would like to finance a part of their purchase with a loan from the broker. Margins are security deposits that can be in the form of cash or some high-quality security. Margin account holders must start with an initial margin. When the account balance falls below the maintenance margin level, the broker issues a margin call, at which time the account holder must come up with enough cash to restore

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the account balance to the maintenance margin level. Short sellers are always required to open margin accounts and must keep the entire sale proceeds plus some additional margin with the broker.

3.11

Cases

Martingale Asset Management L. p. in 2008: 130/30 Funds and a Low Volatility Strategy (Harvard Business School Case 209047-PDF-ENG). The case discusses

the mechanics and the economic implications of leverage and short selling for investment strategies and evaluates minimum volatility stock investment strategies and quantitative investing in general. Deutsche Borse: (Harvard Business School Case 204008-PDF-ENG). The case

explores the implications of Deutsche Borse’s acquisition of a stake in a company specializing in clearing, settlement, and custody of securities across borders. ICEX: Making a Market in Iceland (Harvard Business

School Case 106038PDFENG). The case examines the impact of increased performance on the international visibility and positioning of the Icelandic Stock Exchange and considers various options for stock exchange growth in the backdrop of the country’s strong economic performance during the period.

3.12

Questions and Problems

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3.1. Explain the difference between a stock and a bond. 3.2. Explain the difference between an exchange and an OTC market. 3.3. Explain the difference between a broker and a dealer. 3.4. Explain the difference between a bid and an ask price. 3.5. Explain the difference between a market order and a limit order. 3.6. A forward market is trading for future purchase of a commodity, while a spot

market is trading for immediate purchase. Is the stock market a spot or forward market? 3.7. For the stock market, can there be a difference between the total number of

shares issued by a company and the total number of shares held by investors in the market? If yes, explain why. 3.8. Explain the difference between execution and settlement. 3.9. Explain the difference between bull and bear markets. 3.10. The SEC regulates American stock markets. However, NYSE members have

committees that carry out a host of self-regulatory activities. NYSE members are profit seeking—why would they self-regulate themselves, when regulations only raise their cost of doing business? 3.11. Why does a dealer offer to trade only a fixed amount at the bid and ask prices?

QUESTIONS AND PROBLEMS

3.12. You are a dealer and post a price of $50.00 to $50.50 for a stock. The buy

orders outweigh sell orders, and your inventory is dwindling. How should you adjust the bid and the ask prices, and why? 3.13. What is the difference between an arbitrageur, a hedger, and a speculator? 3.14. What does trading on the OTC mean? 3.15. What is the difference between a stock trading ex-dividend and cum-dividend? 3.16. Suppose that a stock pay a $5 dividend at time t. The dividend is announced

at time (t – 1), when the stock trades cum-dividend at a price of $100. What should be the ex-dividend price at time t? Explain your answer. 3.17. Explain how to sell a stock short, assuming that you do not own the underlying

stock. Why would one short sell a stock? 3.18. Consider the following data: YBM’s stock price is $100. The initial margin is

50 percent, and the maintenance margin is 25 percent. If you buy two hundred shares borrowing 50 percent ($10,000) from the broker, at what stock price will you receive a margin call? 3.19. If the stock price is $105 and the company had paid in the previous year two

quarterly dividends of $0.50 each and two more of $0.55 each, then what is the dividend yield? 3.20. Consider an asset that pays a continuously compounded dividend yield of

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𝛿 = 0.05 per year, which is reinvested back in the asset. If you invest one unit in the asset, how many units would you have after 1.5 years?

71

4 Forwards and Futures 4.1 Introduction 4.2 Forward Contracts 4.3 The Over-the-Counter Market for Trading Forwards Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

4.4 Futures Contracts 4.5 Exchange Trading of a Futures Contract

EXTENSION 4.1 Futures Exchanges in China and India

4.6 Hedging with Forwards and Futures 4.7 Summary 4.8 Cases 4.9 Questions and Problems

FORWARD CONTRACTS

4.1

Introduction

Surprisingly although many think of derivatives as new and sophisticated securities, this is hardly the case. Forward contracts are the oldest known derivatives, tracing their origins back to India (2000 BC), to ancient Babylonia (1894–1595 BC), and even to Roman merchants trading grains with Egypt.1 Futures contracts are “the new kids on the block.” The oldest futures contracts traded in Amsterdam, Netherlands, in the middle of the sixteenth century and on the first futures exchange, the Dojima Rice Exchange in Osaka, Japan, in 1688. As such, there must be good reason why these contracts have existed for millennia to facilitate trade across time. With respect to those who believe that derivatives are new innovations, we see that their timing is only off by three or four thousand years—but what are a few thousand years among friends? We start our study with the simplest and the most basic derivatives, forwards and futures. First we introduce forward contracts that trade in the over-the-counter market. Next we discuss futures contracts. Forward and futures contracts are fraternal twins, for the two derivatives are very similar but not identical. We explore their similarities and differences, then we take a bird’s-eye view of how a company uses forward (and futures) contracts for hedging input and output price risk, a topic discussed in greater detail in Chapter 13.

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4.2

Forward Contracts

Intuitively, a forward contract locks in a price today for a future transaction—contract now, transact later is the mantra. More formally, a forward contract or a forward is a binding agreement between a buyer and a seller to trade some commodity at a fixed price at a later date. This fixed price is called the forward price (or the delivery price, usually denoted by F), and the later date is the delivery date (or the maturity date, usually referred to as time T). Forward contracts are derivatives as their values are derived from the spot price of some underlying commodity. By market convention, no money changes hands when these contracts are created. Such an exchange can only happen if both sides are happy with the terms of the contract and believe the contract is fair, that is, it has a zero value. To understand why, suppose the forward had a positive value to the buyer of the commodity. Then it must have a negative value to the seller. This means that when the seller enters the forward contract, his wealth immediately declines. He has been hoodwinked by the buyer! A rational seller would not freely enter into such a contract. A similar argument holds in reverse if the forward contract has a negative value. Consequently, if no money changes hands when the contract is created, it must have a zero value. As such, the delivery price written into the contract must be a fair price for future trading of the asset. How do we find this delivery price? It’s actually quite straight-forward. Chapters 11 and 12 will show how to find this price using some basic economic principles.

1

See Chapter 8 for an in-depth history.

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Derivatives trade on an exchange or in an over-the-counter (OTC) market. Like a magician pulling a hare out of his hat, derivatives also get created out of thin air! Indeed, they are created when one party wants to trade a particular derivative, finds another party to take the other side of the transaction, negotiates a price, and completes the deal. This observation highlights a classic characteristic of derivatives— they trade in zero net supply markets. A derivative does not exist until a trade takes place—unlike stocks, the buyer’s and seller’s positions net out, leaving the net supply of derivatives at zero. As such, trading derivatives is also a zero sum game. What one side of the contract gains, the other side loses, and vice versa. Some market jargon is useful in understanding derivative transactions. If you agree to buy at a future date, then you take a long position or you are going long. If you agree to sell at a future date, then you are taking a short position or you are going short. A forward buyer is bullish while a forward seller is bearish about the direction of the price of the underlying commodity. Consider the following example of a forward contract.

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EXAMPLE 4.1: The Life of a Forward Contract ■

Today is January 1. Ms. Longina Long longs to buy some gold at a fixed price in the middle of the year. Forwards trade in the OTC (interbank) market, where Long’s broker finds Mr. Shorty Short. Long and Short agree to trade fifty ounces of gold at a forward/delivery price of $1,000 per ounce on June 30 at a mutually convenient place (see Figure 4.1).

■

When the contract begins, its delivery price is adjusted so that the contract is executed without exchanging cash. Consequently, the delivery price is fair, and the market value of the contract is zero at the start.

■

Thus a derivative (forward contract) gets created through mutual agreement. The brokers collect commissions from the traders for their matching service.

■

If Long and Short want, they can close out their positions early through mutual agreement. Theoretically speaking, they can sell their respective sides of the trade to other investors in the secondary market. Practically speaking, it’s a hard task. Surely they can trade with others to stop price risk, but credit risk will remain. If they don’t close out their positions early, Long and Short meet on the maturity date of June 30.

■

What happens on June 30 depends on where the spot settles on that fateful day. Because they agreed on physical delivery, Short sells fifty ounces of gold to Long at $1,000 per ounce. Had they decided on cash settlement instead, the loser would have paid the price difference to the winner. If the spot price of gold is higher than the delivery price of $1,000 per ounce, then Long wins and Short loses— Long pays less for gold than it’s worth in the spot market. Conversely, if the spot price of gold is lower than $1,000 per ounce on June 30, then Long loses and Short wins—she pays more for that gold than it’s worth in the spot market. The forward contract terminates after delivery.

FORWARD CONTRACTS

FIGURE 4.1: Timeline for a Forward Contract

Now (start date) (Time 0 = January 1)

Intermediate dates

Long agrees to buy 50 Traders can close out ounces of gold on June 30, positions by making a which Short agrees to sell. reverse trade (sell if long, buy if short).

Long buys gold for $1,000 from Short. If gold price >$1,000, Long wins. If gold price F and pays F – S(T) if S(T) < F: Short’s payoff at the delivery date T = − [S (T) − F] = F − S (T)

(4.2)

With physical delivery, the short gets paid the delivery price F, a cash inflow. He surrenders to the long an asset that’s worth S(T) in the market. 2

For simplicity, we ignore market imperfections like transaction costs, taxes, and convenience yields from holding a long position in the underlying asset. We will include these market imperfections in later chapters.

FORWARD CONTRACTS

If the contract is cash settled, then the short gets F – S(T) if F > S(T); otherwise, she pays S(T) – F to the long. Notice that the long’s and short’s payoffs are exactly equal and opposite. As mentioned previously, forward contracts are zero-sum games—if you add up the two payoffs, the net result is zero.

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EXAMPLE 4.2: Payoffs to a Long and Short Forward Contract ■

We revisit Example 4.1 again, but this time using notation. Today is January 1 (date 0). The forward price on which Ms. Long and Mr. Short agree for delivery on June 30 (date T) is F(0, T) = F = $1,000 (all prices are for an ounce of gold).

■

Suppose the spot price on the delivery date T is S(T) = $1,005. Long’s forward payoff on delivery is (1,005 – 1,000) = $5 or a gain of $5. Long pays $1,000 for gold worth $1,005 in the spot market. Short’s forward payoff at time T is –(1,005 – 1,000) = − $5 or a loss of $5. The contract forces short to accept this below-market price for gold on the delivery date.

■

If, instead, S(T) is $990 on the delivery date, then Long’s payoff at delivery is (990 – 1,000) = – $10 or a loss of $10. She pays $1,000 for gold worth only $990 in the market. Short’s payoff at delivery is – (990 – 1,000) = $10 or a gain of $10. Observe that for each possible spot price S(T), the long’s and the short’s payoffs add to zero, confirming the zero-sum nature of this trade.

■

The profit and loss for long’s and short’s forward positions on the delivery date (time T) are graphed on the profit diagram in Figure 4.2. The x axis plots the spot price of gold S(T) on the delivery date, while the y axis plots the profit or loss from the forward contract.

■

The payoff to Long is a dashed straight line, making a 45 degree angle with the horizontal axis and cutting the vertical axis at a negative $1,000, which is the maximum loss. This happens when the spot (gold) is worthless, but Long still has to pay $1,000 for it. Above this, each dollar increase in the spot price cuts Long’s loss by a dollar. If gold is worth $1, then Long’s loss is 1,000 – 1 = $999. If it’s worth $800, then the loss is $200. Ms. Long and Mr. Short break even at $1,000. If the gold price ends up higher than $1,000 on the delivery date, then Long sees profits. Figure 4.2 shows that the profit potential for a long forward position is unbounded above.

■

The payoff to Short is the downward sloping dashed straight line, making a negative 45 degree angle with the horizontal axis. Considering the horizontal axis as a mirror, Short’s payoff is the mirror image of Long’s payoff. When gold is worth 0, Short makes $1,000, as worthless gold is sold for $1,000. Thus Short’s payoff line touches the vertical axis at $1,000. For each dollar increase in gold’s price, Short’s profit declines by a dollar. If gold is worth $1, then Short’s profit will be 1,000 – 1 = $999, and so on. If gold increases beyond $1,000, then Short starts losing money. Short’s maximum loss is unbounded— if gold soars, Short plunges into the depths of despair! This shows the risks of naked short selling (short selling without holding an offsetting position in the underlying asset).

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FIGURE 4.2: Profit Diagram for Forward Contract on the Delivery Date Profit Long forward

$1,000

45°

0 F = $1,000

Short forward

-$1,000

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4.3

S(T), spot price on the delivery date

The Over-the-Counter Market for Trading Forwards

The OTC derivatives market has seen tremendous growth over the last four decades. Gigantic markets exist for foreign exchange (“forex”) forwards, money market instruments, and swaps. This is a market for the “big guys”—the banks—with excellent credit ratings. As such, it is often called the interbank market. The participants trade by telephone, telex, and computer, and they tend to be located in the world’s major financial centers such as London, New York, and Tokyo. Example 4.3 shows a typical transaction in the forex (foreign exchange) forward market.

EXAMPLE 4.3: Trading Currency Forward

The Contract ■

Suppose a US company has bought a machine worth €3 million from a German manufacturer with payment due in three months. The treasurer of the US company feels that at $1.4900, the euro is attractively priced in the spot market.3 But he doesn’t know what will happen in three months time; for example, if a euro costs $1.6000, his company will pay an extra $330,000.

THE OVER-THE-COUNTER MARKET FOR TRADING FORWARDS

■

The treasurer would like to pay today, but the company is short of cash. He checks the forward market and finds that DeutscheUSA (a fictitious name), a large commercial bank, bids euros for $1.5000 and offers euros for $1.5010 in three months’ time (see Figure 4.3). He readily agrees and locks in a price of $1.5010 × 3,000,000 = $4,503,000 for the machine. This forward market trade allows the treasurer to eliminate exchange rate risk from the transaction so that the business can focus on its core competency.

The Dealer’s Hedge ■

How does DeutscheUSA protect itself? Having far-flung operations and diverse customer bases, big banks can often find another firm to take the other side of the transaction. Suppose DeutscheUSA has branches in Germany. Contacting its customer base, DeutscheUSA finds a German importer hoping to buy €3 million worth of computer parts from the United States in three months’ time. To shed foreign exchange risk, the German importer agrees to buy €3 million for $1.5000 × 3,000,000 = $4,500,000.

■

DeutscheUSA has done something cool. It has removed price risk for both the US and German importers and managed to earn a riskless spread of (1.5010 – 1.5000) × 3,000,000 = $3,000 in the process! Skillful dealers try to perfectly offset such OTC trades.

■

In reality, such perfect offsets are unlikely, and DeutscheUSA may have residual price risk after netting out all of its foreign currency transactions. Moreover, DeutscheUSA has some credit risk if any counterparty fails, and it will have to keep tab of the transactions until the deal is done. If it desires, DeutscheUSA may hedge any residual net exposure with forex futures or by entering into a forward transaction with another dealer in the forex market.

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3

The International Organization for Standardization, or the ISO, has established three-letter codes for representation of currencies and funds. ISO’s currency codes are USD for the US dollar, AUD for the Australian dollar, EUR for the euro, JPY for the Japanese yen, GBP for the UK pound sterling, etc. When no confusion arises, we will use commonly used expressions (and symbols) like dollar ($), euro (€), yen (¥), and pound sterling or pound (£).

OTC market participants may be classified as brokers, dealers, and their clients. Usual clients are banks, corporations, mutual funds, hedge funds, insurance companies, and other institutions. Many banks act as dealers and make markets in a variety of derivatives. Sometimes clients call up dealers for quotes. Typically a dealer’s sales force makes regular cold calls to potential and existing clients, trying to sell their derivatives products. Owing to competition, simple derivatives like currency forwards aren’t very lucrative. Banks’ proprietary trading desks make more profits when they can identify some special client need and create a customized product. Big players have trading or dealing rooms, where trading desks dedicated to spot, forward, and options trading are located. Forward prices are quoted in terms of a bid and ask, and because there are no specified delivery months like June and September, forward prices get quoted for delivery in one, two, three, six, or more months from the current date. Recently, owing to new financial regulations, many banks are deemphasizing proprietary trading and focusing more on their dealership businesses.

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FIGURE 4.3: Intermediary’s Role in the Forward Transaction Initially

U.S. Co. needs to buy German machines worth €3 million in three months Faces currency risk

Faces currency risk German importer needs to buy computer parts from U.S. in three months

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After DeutscheUSA Enters

U.S. Co. agrees to buy €3 millions in three months German importer needs dollars and agrees to sell €3 million in three months

$1.5010 per euro

$1.5000 per euro

Deutsche USA (dealer) Seller (Short) Buyer (Long)

Deutsche USA has perfectly hedged its book and earns $3,000 spread. It has credit risk but no price risk.

FUTURES CONTRACTS

4.4

Futures Contracts

Forward and futures contracts are fraternal twins. Both involve an exchange of cash for an asset purchased at a later date but at prices negotiated at the start of the contract. Yet their differences (explored in Table 4.2 and Figure 4.4) make it hard to label them as identical. Just as a variety of features like air bags, antilock brakes, power steering, and seat belts make a car easier and safer to drive, a number of added features make futures easier and safer to use. You may view futures as a standardized forward or a forward as a customized futures. Unlike a forward contract, a futures trades on an organized exchange, which is a regulated marketplace where buyers and sellers gather to trade a “homogeneous product.” The federal Commodity Futures Trading Commission and the industry’s National Futures Association regulate commodity exchanges in the United States. In contrast, a forward trades in the OTC market, where no regulator tells them the dos and don’ts. An exchange standardizes futures contracts for easy trading. A trader only has to tell how many contracts to buy or sell and at what price. Standardization helps create a secondary market. It reduces transaction costs and makes futures markets more liquid. A short can easily close out her position before delivery by going long the same contract with another trader, who becomes the new short, while the original long

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TABLE 4.2: Comparison of Futures and Forward Contracts Futures

Forward

a.

Largely unregulated

Regulated

b. Trades in an organized exchange

Trades in an OTC market

c.

Customized

Standardized

d. Usually liquid and has a secondary market

Illiquid and has virtually no secondary market

e.

A range of delivery dates

Usually has a single delivery date

f.

Usually closed out before maturity to avoid taking physical delivery

Usually ends in physical delivery or cash settlement

g. Guaranteed by a clearinghouse and has no credit (counterparty) risk

No such guarantee and has counterparty credit risk

h. Trading among strangers; individual’s creditworthiness is irrelevant

Forward traders know each other and usually have high credit rating or require collateral

i.

Requires margins (security deposits)

No margin requirements (but collateral may be required)

j.

Small transaction costs

Transaction costs are high

k. Settled daily

Settled at maturity

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FIGURE 4.4: A Diagram of a Futures Contract

CFTC, NFA regulate

Broker, Exchange, Clearinghouse monitor

…

…

Daily margin payments throughout contract life

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… Starting date

…

…

…

…

Intermediate dates

…

…

… Delivery/maturity period

position remains undisturbed. In fact, most futures are closed out before delivery. This happens because making physical delivery is a costly exercise, and closing out futures early avoids this expense. If, for some reason, a position remains open until maturity, then delivery must take place on one of several dates during the delivery period. The details of this delivery procedure are discussed in Chapter 9. Contrast this with a forward contract, which is privately negotiated between two traders. As such, the costs of transacting are high. Early termination requires the consent of both sides—you cannot unilaterally close out a position before delivery. Consequently, forwards are highly illiquid, have virtually no secondary market, and are usually held until maturity. Conversely, forwards are tailor-made to suit counterparties’ needs and objectives. Forwards can be designed and traded even in situations when no futures contracts are available.

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EXCHANGE TRADING OF A FUTURES CONTRACT

A futures exchange has associated with it a clearinghouse that clears executed trades and guarantees contract performance by becoming the counterparty to every trader. As such, it eliminates counterparty risk in the transaction. The clearinghouse controls its counterparty risk by requiring traders to keep a margin (security) deposit in their brokerage accounts, which are marked to market and settled at the end of each trading day. We discuss this in greater detail in Chapter 9. By contrast, a forward contract fixes a price for a future transaction that remains locked in over the contract’s life, and settlement only takes place on the delivery date. Futures are primarily designed as risk management tools. They are poor instruments for buying and selling the commodity owing to the hassle of shipping to and from delivery points in specific locations like Chicago and New York, which may ill suit most traders. The US law takes this into consideration and distinguishes contracts on how they end. If delivery is planned and regularly happens, then the contract is likely to be classified as a forward; otherwise, it is a futures. During the 1960s and the early 1970s, some brokerage firms were offering off-exchange forward contracts that were standardized like futures (see Edwards and Ma 1992). Regulators tried to bring them under their jurisdiction, but in 1974, the US Congress denied such incursions by defining forwards as deferred delivery contracts that naturally end in delivery, thereby permitting them to trade away from the exchange floor. Although institutional differences are important, forwards and futures are also different from an economic perspective because a futures contract has daily cash flows, whereas a forward only has a final cash flow. This payout feature implies that futures and forward prices are usually different, but more important, it implies that the risks from holding the contracts differ. Futures face cash flow reinvestment risk with changing interest rates, whereas forwards do not. Because interest rates are random and constantly changing, this reinvestment risk is important, and it affects both pricing and hedging considerations.

4.5

Exchange Trading of a Futures Contract

Next we sketch the trading of a futures contract, which has evolved over time to become more electronic, which efficiently serves user needs. Moreover, this will also help us to understand options exchanges, which evolved from futures exchanges.

EXAMPLE 4.4: Exchange Trading of Futures Contracts

Order Placement ■

Today is January 1. Ms. Longina Long and Mr. Shorty Short are trying to do the transaction of Example 4.1 with exchange-traded futures contracts. They both want to trade fifty ounces of gold at a fixed price on June 30.

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■

Gold futures trade on the NYMEX and the CBOT, both divisions of the CME Group, and on other exchanges. Each regular contract is for one hundred troy ounces of gold, but there are also some mini-contracts of smaller size. No gold futures contract matures in June. There are contracts for April and July, but these months may not suit Long’s buying needs or Short’s selling desire. Standardization allows quick trading, but there is a trade-off in that it reduces available choices.

■

Futures traders must first open margin accounts (security deposits in the form of cash or some acceptable securities) with a Futures Commission Merchant (FCM).4 Old-fashioned Shorty calls up his FCM and dictates a market order to sell one July gold futures at the best available price.

■

Technologically savvy Longina submits her market order through an FCM’s website. Futures traders can place their orders in many other ways besides market orders (see Example 5.8 of Chapter 5 for different order placement strategies).

Trade Execution ■

Shorty’s broker watching his computer screen sees that most trades are occurring at a little over $1,000. He senses the market and submits a limit order to sell one contract at $1,001 or higher trade. The trade is executed.

■

Long and Short pay their respective broker commissions on a round-trip basis. There is no charge when entering a position, but the full charge is due at the time the futures transaction is closed. The amount of commission varies from trader to trader. It could be as low as $10 per round trip trade for a discount broker but far more for a full-service brokerage firm.

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Clearing a Futures Trade ■

In traditional exchanges, before the market reopens, representatives of the clearing members take the trade records to the exchange’s clearinghouse. The clearinghouse checks the records, and once the records match, the clearinghouse clears the trade by recognizing and recording it.

■

The clearinghouse also guarantees contract performance by acting as a seller to every buyer and as a buyer to every seller (see Figure 4.5). The clearinghouse minimizes default risk by keeping margins for exchange members, who, in turn, keep margin for their customers.

Settling a Futures Trade ■

A stock trade is settled when a buyer pays cash and gets the stock’s ownership rights from the seller. In contrast, a futures trade has a special feature called daily settlement, by which profits and losses are paid out daily. At the end of each trading day, a futures position is marked to market by crediting or debiting the day’s gain or loss, respectively, to a trader’s margin account, an amount that equals the difference between yesterday’s and today’s futures prices. If the futures price goes up, Long wins; if it goes down, Short gains.

■

Marking-to-market resets the delivery price at maturity for outstanding futures contracts to the current futures price in freshly minted futures. This process of marking-to-market also resets the value of a futures contract to zero at the end of each trading day.

■

To understand marking-to-market, suppose Long buys one hundred ounces of gold from Short in July for $1,001 per ounce. Suppose the next day’s settlement price is $1,004. Long would screech if you tell

EXCHANGE TRADING OF A FUTURES CONTRACT

her that she now has to buy gold in July at $1,004 because she thought she locked a price of $1,001! However, she would agree to buy at $1,004 if you were to pay her the price difference between her earlier price and the new one: ($1,004 − $1,001) = $3 per ounce. To handle this payment, $3 × 100 = $300 is taken from Short’s margin account and deposited into Long’s margin account. Wouldn’t $300 earn daily interest in a bank account? And wouldn’t such daily receipts or payments happen in an unpredictable fashion? The answer to both questions is yes. As noted earlier, these observations are what differentiate futures from forward contracts. Chapter 9 explains how marking-to-market works in greater detail.

FIGURE 4.6: Role of a Clearinghouse in a Derivative Transaction 301.56........301. .......103.71..... 99.02.......95.01.

Clearinghouse

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Buyer (Long)

Seller (Short)

Buyer (Long)

Seller (Short)

Derivative trades are executed in an exchange where the buyer and seller meet through brokers. After a trade is cleared, the clearinghouse becomes a seller to the buyer and a buyer to the seller. It has counterparty credit risk but no price risk.

Closing the Contract ■

A month later, Longina sells one July gold futures to Miss Tallmadge, closes out her position, and clears her slate. Her effective selling price after all the marking-to-markets is close to $1,001. Taxes, interest charges on margin balances, and other market imperfections will make the realized price a bit different from $1,001. Tallmadge is the new long vis-à-vis Shorty’s short.

■

Shorty decides to physically deliver the gold during the contract’s delivery period in July. He delivers one hundred ounces of gold as per exchange requirements and collects an amount close to $100,100 from Tallmadge.

4

A FCM provides a one-stop service for all aspects of futures trading: it solicits trades, takes futures orders, accepts payments from customers, extends credit to clients, holds margin deposits, documents trades, and keeps track of accounts and trading records.

Futures trading is becoming increasingly popular in different parts of the world, including in countries that were previously hostile to it. Extension 4.1 discusses futures trading in some developing countries, with particular focus on China and India.

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EXTENSION 4.1: Futures Exchanges in China and India

Moving Commodities from Farms to Consumers When you read Grimm’s fairy tales about the old days, you come across farmers who keep a portion of the crop for family consumption and sell the rest in the market. Even today, this is true in many parts of the world, where farmers’ markets are held on a regular basis. But when food moves to towns and cities, you need a supply chain—a network of storage facilities, transporters, distributors, and retailers that take food from the farmer and deliver it to the consumer. Advanced economies use sophisticated technology at each stage of the supply chain to minimize costs and maintain product quality. For example, fruits and vegetables are flash frozen, sent in refrigerated trucks, and kept in temperature-controlled warehouses to extend their shelf life. Owing to this process, restaurants can offer an incredible variety of “fish, flesh, and fowl” such as Maine lobster, Alaskan king crab, Hawaiian mahimahi, and Russian caviar. And as Walmart and many other companies have demonstrated, sophisticated supply chains can significantly lower costs that in competitive markets get passed on to consumers. However, the story is different when commodities move from rural to urban areas in some of the poorest countries. Farmers sell their excess crops to a middleman, who acts as a broker or dealer. The produce goes through several middlemen before it reaches retail outlets. Significant waste can occur during repackaging and reinspection by the middlemen, and poor handling and lack of refrigeration can further diminish the quantity. The farmers get only a small fraction of the final price. True, these intermediaries serve a useful economic function, but they are relevant only because the infrastructure is primitive and sophisticated risk management tools are absent.

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Futures Trading in China Consider China, a country home to over 1.3 billion people. During the 1980s and 1990s, China became interested in developing sophisticated financial markets and institutions. Futures trading began in China in 1993. However, during the first few years, “new exchanges opened with wild abandon, and speculative volume ballooned” (Qin and Ronalds, 2005). Eventually, the Chinese authorities closed more than forty of these exchanges. The Dalian Commodity Exchange, the Shanghai Futures Exchange, the Zhengzhou Commodity Exchange, and the China Financial Futures Exchange (founded in 2006) are the only operating futures exchanges remaining in China today. These exchanges are either fully electronic or use a combination of open-outcry and electronic trading. Interestingly, an October 12, 2009, article in the Wall Street Journal titled “China Targets Commodity Prices by Stepping Into Futures Markets” reported that the leaders of Communist China are planning to use futures exchanges “to fight back” foreign suppliers who “inflate [China’s] commodity prices.” By developing the three commodities exchanges as “major players in setting world prices for metal, energy and farm commodities,” China expects to be less susceptible to exchange prices elsewhere. Futures traders joke that “China is second only to the weather in driving some commodity prices—but less predictable.” Chinese futures prices have begun affecting global prices for many key commodities. For example, Chinese demand was a major contributor to huge swings in crude oil prices during 2008–9.

Futures Trading in India At the start of the new millennium, India removed a four-decade-long ban on commodity futures trading, allowed resuscitation of moribund exchanges, and approved the opening of new ones. Soon afterward, during 2002–3, three major electronic multicommodity exchanges, the National Multi Commodity Exchange of

EXCHANGE TRADING OF A FUTURES CONTRACT

87

India Ltd., the Multi Commodity Exchange (MCX), and the National Commodity and Derivatives Exchange Limited were formed. In yet another development, September 2010 saw the opening of the United Stock Exchange (USE) of India. Backed by government and private banks as well as corporate houses, the USE currently offers trading in currency futures (on-the-spot exchange rate of Indian rupees against the dollar, euro, pound sterling, and yen) and options (on the US dollar Indian rupee spot rate) but plans to expand its offerings to include interest rate derivatives.5 Commodity futures exchanges have also been founded in many other countries, including those in Latin America, Eastern Europe, and Asia. However, Africa has been slow in adopting futures trading. 5

See www.useindia.com/genindex.php#.

In futures trading, the terms open interest and trading volume need careful distinction: trading volume is the total number of contracts traded, whereas open interest is the total number of all outstanding contracts, which may also be counted as the total number of long (or short) positions. Trading volume is a measure of a market’s liquidity: the more the volume is traded, the more active the market is. In contrast, the open interest is a measure of the market’s outstanding demand for or exposure to a particular commodity at the delivery date. Example 4.5 further explores the distinction between these two concepts.

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EXAMPLE 4.5: Volume versus Open Interest ■

Suppose that a July gold futures just becomes eligible for trading. Heather buys twenty of those contracts from Kyle. Trade records will show the following (for convenience, they are also shown in Table 4.3): - Heather is long twenty contracts. - Kyle is short twenty contracts. - Trading volume is twenty contracts. - Open interest is twenty contracts.

■

Heather decides to reduce her exposure. She sells ten contracts to Tate. As a result, - Heather is now long ten contracts. - Tate is long ten contracts. - Kyle is short twenty contracts, as before. - Trading volume rises to thirty contracts. - Open interest is twenty contracts, as before.

■

Kyle reduces his short position. He buys five contracts from Heather. Consequently, - After selling five contracts, Heather is long five contracts. - Tate is long ten contracts, as before. - Kyle is short fifteen contracts after this trade. - Trading volume, which adds up the number of trades, is thirty-five contracts. - Open interest, which is the sum total of all outstanding long positions (or short positions), is now fifteen contracts.

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TABLE 4.3: Trade Records for Gold Futures Contracts Kyle sells twenty contracts to Heather

Trader Heather

Long

Short

Trading Volume

20

Kyle

20

Open Interest 20

20

Heather sells ten contracts to Tate

Trader Heather

Long

Trading Volume

10

30

Open Interest 20

20

Kyle Tate

Short

10

Heather sells five contracts to Kyle

Trader Heather

Long 5

Kyle

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Tate

4.6

Short

Trading Volume 35

Open Interest 15

15 10

Hedging with Forwards and Futures

Forwards and futures are widely used for hedging commodity price risks at future dates. Consider a typical firm, which uses inputs to produce one or more outputs. Input and output prices are susceptible to price fluctuations. Usually, a firm tries to shed price risk for both its inputs and outputs, provided it is not costly to do so. Then it can concentrate on production; in other words, it can focus on what it does best. Obviously, it is a judgment call to decide which risks to cure and which to endure. Suppose that to reduce input price risk, a firm sets up a long hedge (or a buying hedge) by taking a long position in a forward contract. For example, a company that mines and refines gold in some remote area and uses natural gas to power its electricity generators may find it prudent to buy a forward contract on natural gas. If natural gas prices go up in the future, then the company pays more in the spot market, but the spot market loss is largely offset by the gain on the forward contract. On the flip side, if natural gas prices go down, the company will surrender some of its spot market gains through losses on the forward position. This is always the case when one employs a hedging strategy. Hedging with forwards (and futures) cuts both ways: it

SUMMARY

reduces the risk of financial loss from adverse price movements, but also takes away potential gains from favorable price changes. Alternatively, to reduce output price risk, the company can establish a short hedge (or selling hedge) by taking a short position in a forward contract. If the company sells gold forward, then it will remove or lessen the potential for loss (as well as gains) from future spot price fluctuations. If the spot goes down, then the company loses money when it sells gold in the cash market, but it profits from the forward contract. The losses and gains are reversed when the spot price goes up. Hedging is analogous to purchasing an insurance policy on the commodity’s price. It pays off when prices move in an adverse direction. But as with all insurance policies, there is a cost—the premium. If the price does not move in an adverse way, you paid for insurance that you didn’t use. This is the cost. Risk-averse individuals will often buy insurance despite this cost, and analogously, firms will often hedge their input or output price risk.

4.7

Summary

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1. The buyer and seller in a forward contract agree to trade a commodity on some later delivery date at a fixed delivery (forward) price. Forwards are zero net supply contracts. Forward trading is a zero-sum game. On the maturity date, the buyer’s (or the long’s) payoff is the spot price minus the delivery price. The seller’s (or the short’s) payoff is equal in magnitude but opposite in sign to the long’s payoff. 2. There are organized OTC (interbank) markets for trading forwards, which are dominated by banks and other institutional traders. These markets are huge. For example, foreign currency forwards attract billions of dollars of trade every day and have outstanding obligations measured in trillions of dollars. 3. The buyer and seller of a futures contract agree to trade a commodity at a fixed delivery price on the maturity date. A futures contract is similar to a forward. The long position holder in both these contracts agrees to buy the commodity, while the short agrees to sell. The forward or futures price is set so that no cash changes hands when the contract is created. This implies that the contracts have zero initial value. 4. Exchange-traded derivatives are highly regulated contracts that trade in organized exchanges. A clearinghouse clears trades and guarantees contract performance. Traders are required to keep margins (security deposits) that are adjusted daily to minimize default risk. Exchanges also act as secondary markets where traders can take their profits or cut their losses and exit the market. 5. Many firms reduce price risk by trading futures and forward contracts. A firm can lower input price risk by setting up a long hedge buying a forward and reduce output price risk by establishing a short hedge selling a forward.

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4.8

Cases

The Dojima Rice Market and the Origins of Futures trading (Harvard Business

School Case 709044-PDF-ENG). The case blends business history with policy issues surrounding the introduction of rice futures at the Dojima Exchange, the world’s first organized (but unsanctioned) futures market. United Grain Growers Ltd. (A) (Harvard Business School Case 201015-PDFENG).

The case considers how a Canadian grain distributor can identify and manage various risks. ITC eChoupal Initiative (Harvard Business School Case 297014-PDF-ENG). The

case discusses the use of Internet technologies and derivative contracts to help poor farmers in rural India.

4.9

Questions and Problems

4.1. Define a forward contract. If a forward contract on gold is negotiated at a

forward price of $1,487 per ounce, what would be the payoff on the maturity date to the buyer if the gold price is $1,518 per ounce and to the seller if the gold price is $1,612 per ounce? 4.2. Define a futures contract. 4.3. Discuss the similarities and differences between forward and futures contracts. 4.4. What are the costs and benefits to a corn grower trading a forward contract? If

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she is expecting a harvest in three months, should she buy or sell the derivative? 4.5. Which contract has more counterparty risk, a forward contract or a futures

contract? Explain your answer. 4.6. Discuss the benefits of standardization of a futures contract. 4.7. What are the two roles that a clearinghouse plays in the case of a futures

contract? 4.8. Ang, Bong, Chong, and Dong are trading futures contracts. Carefully identify

the trading volume and open interest for each trader from the following transactions: a. Ang buys five October silver futures from Bong. b. Chong sells nine December silver futures to Bong. c. Dong buys ten October silver futures from Chong. d. Bong sells five December silver futures to Ang. e. Ang sells three October silver futures to Chong. 4.9. Suppose you are a trader specializing in futures on corn, wheat, oats, barley,

and other agricultural commodities. From the following list, which risks do you face (mark each with a yes or no): credit risk, market risk, liquidity risk, settlement risk, operational risk, legal risk.

QUESTIONS AND PROBLEMS

4.10. For forward and futures contracts, what is the difference between physical

delivery and cash settlement? 4.11. Why does a futures contract have zero value when it is first written? 4.12. What is marking-to-market for a futures? Why is this marking-to-market

important for reducing counterparty risk? 4.13. Explain why a futures contract is a zero-sum game between the long and short

positions. 4.14. Are forward contracts new to financial markets? Explain. 4.15. Can you think of a reason why forward prices and futures prices on otherwise

identical forward and futures contracts might not be equal? 4.16. What is the OTC market for trading derivatives? How do OTC markets differ

from exchanges? 4.17. When holding a futures contract long, if you do not want to take delivery of

the underlying asset, what transaction must you perform? Explain. 4.18. If you are short a futures contract, why do you not have to borrow the futures

contract from a third party to do the short sale? 4.19. Is the futures price equal to the value of the futures contract? If not, then what

is the value of a forward contract when it is written? Explain. 4.20. Is the forward price equal to the value of a forward contract? If not, then what

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is the value of a forward contract when it is written? Explain.

91

5 Options

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5.1 Introduction

Price Bounds for American Puts

5.2 Options

5.5 Exchange-Traded Options

5.3 Call Options Call Payoffs and Profit Diagrams

5.6 Longs and Shorts in Different Markets

The Call’s Intrinsic and Time Values

5.7 Order Placement Strategies

Price Bounds for American Calls

5.8 Summary

5.4 Put Options Put Payoffs and Profit Diagrams The Put’s Intrinsic and Time Values

5.9 Cases 5.10 Questions and Problems

OPTIONS

5.1

Introduction

After introducing options in an intermediate finance class, a professor asked, “Is auto insurance an option?” A serious student from the front row replied, “It is not an option because state law requires it, but you have the option to choose from different insurance companies.” Though the student was entirely correct, this was not the answer for which the professor was hoping. He took a deep breath and went on to explain that put options are very similar to insurance contracts because they restore an asset’s value after a decline and that much of option jargon comes from the insurance industry—and he resolved to frame his questions more carefully in the future. As the example illustrates, the word option has many meanings in daily life. Exchange-traded options are well-defined contracts with specific features. This chapter describes these option features and discusses how they trade. At the chapter’s end, we discuss how traders buy or sell options for various strategic objectives.

5.2

Options

Options come in two basic types: calls and puts, names you have heard before. A call option gives the buyer: - the right, but not the obligation - to buy a specified quantity of a financial or real asset from the seller - on or before a fixed future expiration date

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- by paying an exercise price agreed on today. The buyer is also called the holder or the owner. The seller is known as the grantor, the issuer, or the writer. The known future expiration date is also called the maturity date, and the fixed price is known as the exercise price or strike price. The holder exercises the call when she exercises her right and buys the underlying asset by paying the exercise price; otherwise, she lets the call expire worthless. By contrast, a put option gives the right to sell. Specifically, a put option gives the owner the right to sell the asset to the writer at the strike price until the expiration date. A buyer exercises the put when he sells the underlying asset and receives the strike price from the put writer; otherwise, he can let the put expire worthless. Options do not come free—the option buyer must pay the writer a fee (option’s price or premium) for selling those rights. The terms call and put come from what the buyer can do with these options. A call gives the option to buy, that is, to call the asset away from the writer. Conversely, a put gives the option to sell or to put the asset to the writer. In each case, the writer stands ready to take the other side of the buyer’s decision. We say that a call buyer is bullish because she expects the underlying asset to go up in value, and a put buyer is bearish because he expects the asset price to decline. Naturally, their counterparties hold opposite market views: the call writer is bearish and the put writer is bullish. We also say that the option (call or put) buyer is long the option, so the writer ends up with a short position.

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CHAPTER 5: OPTIONS

Notice that calls and puts have a lot in common with insurance policies and may be viewed as insurance contracts for hedging or risk reduction. If someone hits a car, the insurance company pays the owner enough money (subject to some deductibles) to restore the car to its original value. Likewise, a put pays the holder in case of a meltdown in the asset’s value. A call buyer benefits from an asset price rise but does not suffer from its decline. The call buyer pays for insurance to avoid losses if the stock falls. The call writer provides this insurance. For these reasons, an option price is called a premium. A premium is what you pay for the insurance. Calls and puts are examples of plain vanilla or standard derivatives. Later we will consider nonstandard or exotic derivatives, which are derivatives with more complex payoffs. Plain vanilla options are usually defined in one of two ways: American or European. A European option can only be exercised at the maturity date of the option, whereas an American option can be exercised at any time up to and including its expiration date. What we defined earlier were calls and puts of the American type. The adjectives European and American tell how the options differ in their exercise choices and have nothing to do with where they trade—in Europe, America, Africa, Asia, Australia, or even in Antarctica. European-style options trade in the US and across the Atlantic, and American-style options transact in Europe. Both American and European options have the same value on the expiration date, if the American option remains unexercised. Before expiration, however, American options are at least as valuable as their European counterparts. They duplicate what European options can do, but they also offer an early exercise feature, which is more. You don’t pay less when you get more. Chapter 6 will show how the assumption of no arbitrage makes this happen. Such simple insights are useful when trading options. The important question studied in later chapters is, how do you find an option’s premium? This has bewildered academics for a long time. A breakthrough came in 1973 with the development of the seminal Black–Scholes– Merton (BSM) option pricing model (see Chapter 19). The importance of the BSM model can hardly be overstated—it helped spawn the entire field of derivative pricing that we see today. However, option pricing still remains a Herculean task. For example, there is no generalized closed-form solution for finding an American option’s premium.1 This is important because most exchange-traded options are of the American type, and we need to price them. When closed-form solutions are not available, numerical procedures (computer programs) are used instead (see Wilmott 1998; Duffie 2001; Glasserman 2003). There remain many open questions, and option pricing remains an intellectually challenging field to which interesting contributions can still be made.

A closed-form solution is an exact mathematical formula like x = (a + b)2 . If you know a and b, you can easily solve for x. The BSM model solves an option’s price in terms of known variables and functions like the cumulative normal distribution.

1

CALL OPTIONS

5.3

Call Options

Call Payoffs and Profit Diagrams Option payoffs on the expiration date or at early exercise are easier to understand in a perfect market, where market imperfections like transaction costs, taxes, and trading restrictions are assumed away. In reality, the market contains such imperfections. The brokers who match option trades charge commissions for their services. A dealer will quote an ask price (say, $4) and a bid (say, $3.90) for the option premium. Large trades tend to have market impact, and the entire order may not be filled at the price posted by a dealer. Unfortunately, these real-world features detract from understanding the basic payoffs. Hence we study option payoffs under a perfect market structure. Once you know how these contracts work, you can adjust for these imperfections. Example 5.1 considers a call buyer’s and writer’s payoffs and profit diagrams on the expiration date.

EXAMPLE 5.1: European Call Option Payoff and Profit Diagrams

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Call Buyer’s Payoff ■

Today is January 1 (time 0). An April 100 European call option on the stock of Your Beloved Machines Inc. (YBM) is created when Ms. Longina Long pays the premium c = $4 to Mr. Shorty Short, who becomes the writer. This call has a strike price of K = $100, and it ceases trading on the third Friday of April. This is the only day this European call can be exercised (time T); see Figure 5.1 for this option’s timeline.2

■

Long’s exercise decision depends on the stock price S(T) on the fateful expiration date. If S(T) is $106, then Long exercises the call because the stock is worth more than the strike (traders say that the option is in-the-money). She pays the strike, receives the stock worth S(T), and makes [S(T) – K] = $6 in the process. The $4 premium is a sunk cost and does not affect the exercise decision.

■

Conversely, if the stock is worth less than the strike, then the option expires worthless—out-of-themoney, in traders’ parlance. If Long is keen to acquire the stock, she should directly tap the stock market. It’s imprudent to exercise out-of-the-money options. For example, if S(T) is $92, she gets 92 – 100 = –$8 by exercising the call, an outcome she should avoid. And when S(T) equals the strike price, the option is at-the-money, and the long is indifferent regarding exercise. In reality, a comparison of transaction costs for purchasing the stock is likely to influence her decision. For example, suppose Long is planning to purchase a thinly traded stock. If the stock’s spread is too high, Long may exercise an at-the-money call (or even a call that is slightly out-of-the-money) to more cheaply acquire the stock.

■

True to the proverb “a picture says a thousand words,” patterns emerge when we graph these payoffs in a payoff diagram. These diagrams plot the stock price at expiration, S(T), along the horizontal axis and the option payoff (or gross payoff ) along the vertical axis. Long call’s payoff starts at 0 value when the stock price is zero and lies flat along the axis until the stock price reaches K = $100 (see Figure 5.2). Beyond this, the payoff increases at a 45 degree angle, rising by a dollar for each dollar increase in S(T).

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Call Writer’s Payoff ■

Because option trading is a zero-sum game, Short’s call’s payoff is the mirror image across the horizontal axis of Long’s payoff. The writer loses nothing at expiration until S(T) reaches $100, beyond which his payoff declines at a negative 45 degree angle. When S(T) = $106, Short loses $6, which is Long’s gain. When S(T) = $92, Short loses nothing.

Call Buyer’s Profit ■

Payoff diagrams fail to reveal why Short, who has a zero or negative payoff on the expiration date, will get into this venture. This limitation is overcome by a profit diagram, which adds the option’s premium to the trader’s payoff. The profit diagram plots S(T) along the horizontal axis but has profit (or loss) from option trading along the vertical axis. For convenience, we ignore any interest that the option premiums may earn in the margin account. This allows us to combine the cash flows at the option’s initiation and expiration dates on the same diagram.

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FIGURE 5.1: Timeline for a Call Option

Today (start date) (Time 0)

Intermediate dates

Buyer (Long) pays the Long can exercise an call premium and gets American option. the call from the writer (Short) (received as an electronic account). Traders can close out their positions. (If none of these happens, traders meet at time T).

Expiration/maturity date (Time T)

If S(T) > K, long exercises: Long gives up the call, pays the strike price, and gets the stock from the writer. Long nets S(T) – K. If S(T) < – K, call expires worthless.

CALL OPTIONS

■

If S(T) does not exceed the strike price K = $100, Long loses c = $4, the premium she paid for the call. This is her maximum loss. When S(T) exceeds $100, Long will definitely exercise the call. A dollar increase in S(T) raises the value of her position by a dollar and cuts her losses. Consequently, the profit graph is a horizontal straight line at –$4 until it reaches the strike price, where it increases at a 45 degree angle. When S(T) = $104, Long’s $4 gain exactly offsets her initial $4 investment and gives “zero profit.” When S(T) goes beyond $104, it’s even sweeter. Technically speaking, the call holder has limitless profit potential (see Figure 5.2). Focusing on our familiar points, when S(T) = $106, Long has a net gain of $2 because she makes $6 but she paid $4 for the call. Conversely, when S(T) = $92, Long does not exercise and loses only the $4 premium.

FIGURE 5.2: Payoff and Profit Diagrams for an April 100 Call Option on the Expiration Date Payoff

Payoff

Long call 45˚

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0

100

S(T), stock price at expiration

Profit

K

0

–45˚

S(T)

–45˚

S(T)

Short call

Profit

Buy call

0 –4

45˚

100 c+K

S(T)

c 0

104 K

Sell call

Call Writer’s Profit ■

When S(T) does not exceed the strike, the call writer keeps the $4 premium collected up front. This is also the maximum profit. When S(T) increases above $100, Long’s exercise reduces this amount. Zero profits occur when S(T) = $104. Below this level, Short plunges deeper into losses (see Figure 5.2).

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In the example considered, when S(T) = $106, the writer has a loss of $2. When S(T) = $92, Long does not exercise, and Short’s profit is $4. Because option trading is a zero-sum game, the total gain of both traders is zero. In reality, brokerage commission lowers the profits accruing to each trader. ■

Notice what the diagrams reveal: call writing can generate far larger losses than call buying. The maximum loss to the buyer is the premium. In contrast, the writer can literally fall into a bottomless abyss where the maximum loss is unbounded. Recognizing this asymmetry, the options exchange requires a call writer to hold more funds in a margin account than a buyer.

2

The OCC (www.optionsclearing.com/about/publications/expiration-calendar-2010.jsp) and exchange websites label the following Saturday as the expiration date. This is a technicality. The third Friday, which we label as time (date) T, is the relevant economic date for our purpose because it is (1) the day when the option stops trading, (2) the only day a European option can be exercised, and (3) the last day an American option may be exercised. For simplicity, we will use the terms exercise date and expiration date interchangeably.

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The Call’s Intrinsic and Time Values An American call is identical to a European call, except that it can be exercised any time until expiration. Remember that if an American call is not exercised early, then it has the same value as a European call on the expiration date so that the payoff diagrams are identical. And prior to expiration, the exercise strategy for an American option is guided by two facts: (1) one should never exercise an out-of-the-money option, and (2) when exercising an in-the-money call early, one receives the stock price minus the strike price. In other words, if an American call is exercised early, then its payoff looks like that of a European call on the expiration date. We summarize these insights in Result 5.1.

RESULT 5.1 Intrinsic (or Exercise) Value of a Call Option A call holder’s payoff at expiration or if exercised early is Call’s intrinsic value ≡

S − K for S > K {0 otherwise

(5.1)

which may be written as Call’s intrinsic value = max (S − K, 0)

(5.2)

where S is the stock price on the exercise date, K is the strike price, and max means the maximum of the two arguments that follow. This is a call option’s boundary condition. The payoff is also referred to as the call’s intrinsic or exercise value.

CALL OPTIONS

What does this mean? For S < K, the term (S – K) is negative, the call is not exercised, and 0 is the value. When S ≥ K, the call’s value equals S – K. The profit is obtained by subtracting the call’s premium from the preceding payoff. This result has a simple corollary. It implies that if the stock price hits zero, then the call price is also zero. Indeed, if the stock hits zero, it has no future, and it is never going to take a positive value again. Consequently, a call, which gives the right to purchase the stock in the future, must also be zero. The intrinsic value is only part of an option’s premium. The remainder is the additional value accruing to an option because the stock price may increase further before maturity. Consequently, we define the time value of a European or American option as Time value of an option = Option price − Option’s intrinsic value

(5.3)

Time value declines as one approaches the expiration date and (by its definition) becomes zero when the option matures. Example 5.2 computes the time value of a call option.

EXAMPLE 5.2: Time Value of a Call Option

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■

On some date t before expiration, suppose that YBM’s stock price S(t) = $101 and the call price c(t) = $2.50. If the strike price K = $100, then Call option’s time value = Call price − Call’s intrinsic value = c (t) − [S (t) − K] = 2.50 − (101 − 100) = $1.50

Price Bounds for American Calls Although we are yet to price options, the boundary condition and some simple reasoning can determine what call option prices are admissible and what prices aren’t (see Example 5.3).

EXAMPLE 5.3: Price Bounds for an American Call Option ■

As in Example 5.1, consider an American call option with a strike price K = $100. Its price can never exceed the underlying stock’s price. Indeed, the call is a security that gives the holder the right to buy the underlying stock by paying the strike price. If you have to choose between the two—a gift of the stock versus getting the stock by paying some money—which one would you prefer? Getting the gift, of course, which implies that the stock is worth more than the option.

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Consequently, a call’s price can never exceed the underlying stock’s price. Figure 5.3 plots today’s stock price S(t), written as S for simplicity, along the horizontal axis and has both the stock and the call price along the vertical axis. The upward sloping 45 degree line from the origin represents the stock price. The call price can never exceed this value, and the 45 degree line forms an upper bound. A rational investor will never let a call’s price fall below the boundary condition given by Result 5.1, max(S – 100, 0)max(S – 100, 0), which forms a lower bound. This is the kinked line that starts at the origin and increases at a 45 degree angle when the stock price is $100.

FIGURE 5.3: Price Bounds for an American Call Prices Upper bound Lower bound

Call prices lie here 45˚

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0

■

K = $100

S, stock price

Let us illustrate these bounds with some numbers: - When the call is out-of-the-money, say, the stock price is $90, the time value prevents the call price from being zero. So it will lie above the horizontal axis. However, the call price can never exceed the stock price of $90, and hence it will lie below the 45 degree line emanating from the origin. - When the call is in-the-money, say, the stock price is $115, the call is worth at least S – K = 115 – 100 = $15 if exercised. So the call price can never fall below the lower bound, and of course, it can never exceed $115, which lies on the upper bound.

5.4

Put Options

Put Payoffs and Profit Diagrams We illustrate a put option’s payoff and profit diagrams with Example 5.4.

PUT OPTIONS

EXAMPLE 5.4: European Put Option Payoff and Profit Diagrams

Put Option Payoffs ■

■

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■

Today is January 1 (time 0). Ms. Long pays Mr. Short p = $6 and buys a May 50 European put option on Boring Unreliable Gadgets Corp (BUG). This put gives Long the right to sell BUG stocks on the exercise date, the third Friday of May (time T), at the strike price K = $50. A put’s timeline is similar to that for the call given in Figure 5.1. If the put is not closed out early, the traders meet on the expiration date T. If the stock price is less than the strike price (the put is inthe-money), Long exercises the put, receives K dollars, surrenders the stock worth S(T), and makes [K – S(T)] in the process. For example, if S(T) = $40, then Long sells the stock for $50 and makes $10. Conversely, if S(T) is greater than K (the put is out-of-the-money), it’s unwise to exercise. For example, if S(T) = $55, Long gets (50 – 55) = – $5 if she exercises and zero if she doesn’t. Exercise is irrelevant if BUG closes at $50 (the put is at-the-money).3 These considerations enable us to draw the payoff diagrams. For large stock prices, the put holder’s payoff has zero value, and it lies on the horizontal axis. As the stock price declines and dips below the strike, the payoff forms a kink at K and increases at a 45 degree angle, increasing by a dollar for each dollar decrease in the stock price (see Figure 5.4). The profit peaks when the stock is worthless, but Long can sell it for K dollars as per contract terms.

FIGURE 5.4: Payoff and Profit Diagrams for a May 50 Put Option on the Expiration Date Payoff

Payoff

50 Long put 0

45˚ 50

S(T), stock price at expiration

0

K –45˚

S(T)

Short put –50

Profit

Profit

44 Buy put 0 –6

45˚

6 0

K S(T)

44 –45˚ Sell put

–44

S(T)

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Because option trading is a zero-sum game, the put writer’s payoff is the mirror image across the xaxis of the buyer’s payoff. When S(T) is greater than $50, the put seller loses nothing, and the payoff coincides with the horizontal axis. When the stock goes below the strike, the payoff decreases below the horizon line at a 45 degree angle starting at K, and it reaches the nadir when the stock becomes worthless.

Put Option Profit ■

■

■

The profit diagram adjusts the payoff diagram for an option’s premium. If S(T) is greater than the strike price K = $50, Long loses the premium p = $6, which is the most she can lose. When S(T) falls below $50, Long exercises the put—each dollar decrease in S(T) raises the value of her position by a dollar and cuts her losses. Consequently, the profit graph is a horizontal line at – $6 for high values of S(T). It increases upward at a 45 degree angle when S(T) touches K = $50 on its downward journey, and it cuts the horizontal axis at the zero-profit point when S(T) = $44. It hits the vertical axis and makes an intercept of $44 when S(T) = 0 (see Figure 5.4). The profit graph for a put writer is the mirror image of the buyer’s graph across the horizontal axis. Notice that both the maximum profit and the biggest loss are bounded for a put option. Next we illustrate these profit (or loss) computations. For S(T) = $40, Long makes (50 – 40) = $10 by exercising the put. However, he paid $6 for the option, netting a profit of 10 – 6 = $4, which is also Short’s loss. For S(T) = $55, Long does not exercise the put, and her loss is the $6 premium, which is also Short’s profit.

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3

Transaction and liquidity costs may slightly modify this decision. If the long has a substantial holding in the underlying stock that is thinly traded and wants to sell, then she may exercise an at-the-money or slightly out-of-the-money put option to sell the stock at minimum transaction cost.

The Put’s Intrinsic and Time Values An American put may also be exercised before expiration. Of course, if an American put is not exercised early, then the American and European puts have identical value on the expiration date so that their payoff diagrams are identical. Before expiration, the exercise strategy for an American option is determined by two facts: (1) one should never exercise an out-of-the-money option, and (2) in case of early exercise of an in-the-money put, the value is the strike price minus the stock price. When exercised early, an American put’s payoff looks like that of a European put on the expiration date.

PUT OPTIONS

These observations are summarized as Result 5.2.

RESULT 5.2 Intrinsic (or Exercise) Value of a Put Option A put holder’s payoff at expiration or if exercised is Put’s intrinsic value ≡

K − S for S < K {0 otherwise

(5.4)

which may be written as Put’s intrinsic value = max (K − S, 0)

(5.5)

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where K is the strike price, S is the stock price on the exercise or expiration date, and max means the maximum of the two arguments that follow. This is a put option’s boundary condition. The payoff is also referred to as the put’s intrinsic (or exercise) value.

For example, consider a put with a strike price of K = $50. When S(T) = $55, K – S(T) = – $5. As the maximum of –$5 and 0 is clearly 0, the put value p(T) = 0 at expiration. When S(T) = $40, then K – S(T) = $10 is the intrinsic value. For S(T) < K, the positive term [K – S(T)] is the put’s payoff. When S(T) ≥ K, the term K – S(T) is no longer positive, and zero becomes the put’s intrinsic value. Example 5.5 illustrates the computation of a put’s time value.

EXAMPLE 5.5: Time Value of a Put Option ■

At time t, let BUG’s stock price S(t) = $48 and its put price p(t) = $3 for a strike price K = $50. Then Put option’s time value = Put premium − Put’s intrinsic value = p (t) − [K − S (t)] = 3 − (50 − 48) = $1

Price Bounds for American Puts As with calls, we can use the boundary condition to determine price bounds for put options (see Example 5.6).

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EXAMPLE 5.6: Price Bounds for an American Put Option ■

As in Example 5.4, consider an American put option with a strike price of K = $50. Its value can never exceed the strike price because the put gives the holder the right to sell the underlying stock at the strike price. Thus the maximum payoff from the put is the exercise price K = $50. This forms an upper bound for the put price and is depicted by the horizontal line at $50 in Figure 5.5.

■

The put’s price can never fall below max(0, 50 – S), the boundary condition given by (5.3), which forms a lower bound. This is the line that originates at $50 on the vertical axis, decreases at a 45 degree angle until it hits $50 on the horizontal axis, and then continues along the horizontal axis.

■

We illustrate these bounds with an example (see Figure 5.5): - When the put is in-the-money, say, the stock price is $10, the put price cannot fall below its exercise value 50 – 10 = $40. Because the put price cannot rise above its upper bound of $50, it lies between $40 and $50. - When the put is out-of-the-money, say, the stock price is $60, the put price cannot fall below the horizontal axis because no one would exercise and lose money. Of course, it cannot exceed its upper bound: the flat line at $50 above the horizontal axis.

FIGURE 5.5: PRICE BOUNDS FOR AN AMERICAN PUT

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Prices

$50

Upper bound Put prices lie here

0

Lower bound K = $50

Put option prices cannot lie in the shaded regions above the upper bound or below the lower bound.

S, stock price

EXCHANGE-TRADED OPTIONS

5.5

Exchange-Traded Options

Relative to futures contracts, exchange-traded options are newcomers. They started trading in 1973 with the opening of the Chicago Board Options Exchange (CBOE). Today options trading is dominated by big exchange groups, which offer diverse product lines (stocks, bonds, options, futures, etc.). The Options Clearing Corporation (OCC) clears the bulk of exchange-traded options in the US. As such, the OCC becomes the guarantor of contract performance to each buyer and seller. As discussed before, a clearinghouse makes derivatives safer to trade, helps develop a secondary market, allows anonymous trading, and eliminates the need to track one’s counterparty. These benefits have greatly contributed to option market growth. Example 5.7 considers a hypothetical options trade.

EXAMPLE 5.7: Exchange Trading of a Call Option

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Order Placement ■

Today is January 1 (time 0). Ms. Longina Long and Mr. Shorty Short have opened margin accounts and have been approved for trading options with their respective brokers. Unbeknownst to each another, they simultaneously decide to trade ten April 100 call options on Your Beloved Machines Inc. (YBM). These American calls have a strike price of K = $100 and an exercise date of the third Friday of April. Because each option contract is on one hundred shares, ten contracts reflect gains or losses from one thousand shares.

■

Tech-savvy Longina flips open her laptop and logs onto her brokerage account. Finding that YBM April calls have a bid price of $4.00 and an ask–offer price of $4.05, she types in a limit order that instructs her broker to buy ten contracts at $4.00 or a lower price. Old-fashioned Shorty calls up his broker and places a market order that is immediately executed. Small orders are matched via computers and are filled and confirmed within a few seconds. Most option trades are executed on the exchange floor.

■

Suppose Long’s and Short’s orders are sent via computer to their respective brokerage firms.

Trade Execution, Clearing, and Settlement ■

Shorty’s broker accepts Longina’s limit order, and ten contracts are executed at c = $4.00. Longina sees the trade confirmation on her web page, and Shorty gets a confirmatory phone call from the broker’s representative. Vital transaction data also go into the exchange’s main computer system.

■

Commission rates vary. A discount broker typically charges a fixed rate, which may be as small as $5 to $20 per trade, and a variable rate, which typically ranges from 50 cents to $2 for each option on one hundred shares. Many firms charge lower commissions for (1) internet orders; (2) active traders; (3) the simplest orders like market orders; (4) small-sized orders, because they are unlikely to be based on privileged information; and paradoxically, (5) larger orders, to give customers a break. Brokers charge two commissions: one at the time of buying (or selling) and a second commission when selling (or buying back) or exercising the option.

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Suppose Long’s discount broker charges her $10 for the trade plus a dollar per contract—$20 in all. Consequently, Long’s brokerage account is debited an amount equal to the buying price plus commission: Price × 100 × (Number of contracts) + Commission = 4 × 100 × 10 + 20 = $4, 020

■

Shorty has a full-service broker who charges $50 for the trade plus $3 per contract—$80 in all. Short’s margin account is credited an amount equal to the selling price less the commission: Price × 100 × (Number of contracts) − Commission = 4 × 100 × 10 − 80 = $3, 920

Because selling options is a risky transaction, Short’s broker requires him to keep the entire proceeds as well as some extra funds as a security deposit in the margin account.

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Closing the Contract ■

Being exchange-traded contracts, options have secondary markets. Traders retain the flexibility of taking their profits or cutting their losses and walking away. Long may sell the option to a third party and close out her position—she may win or lose in the process. Quite independently, Short can buy the option in a closing transaction, and a third trader then becomes the writer. Because this is an American call, Long can exercise at any time until the maturity date.

■

Fluctuations in option prices affect the margin account’s value. Suppose that YBM rallies tomorrow and the call price increases by $3. Then Long’s margin account gains $3,000, and Short’s margin account declines by the same amount. If his maintenance margin is breeched, Shorty has to add funds to his margin account.

■

Suppose that the option is neither exercised nor closed out early so that the counterparties meet again on the expiration date. To avoid the hassle of paying the broker a fee for exercising and yet another fee for trading YBM shares, they decide to close out their positions just before the option’s expiration and trade at a $7 premium per option. Then Long’s profits are Sale proceeds − Cost = (7 × 100 × 10 − 20) − 4, 020 = $2, 960 and Short’s profit (remember, his brokerage costs are higher) is Sale proceeds − Cost = 3, 920 − (7 × 100 × 10 + 80) = − $3, 160

ORDER PLACEMENT STRATEGIES

5.6

Longs and Shorts in Different Markets

Market practitioners often use the terms long and short to convey the notions of buying and selling. But these terms have different meanings in different markets. For derivatives, long means “buy,” and short means “sell.” Derivatives trade in zero net supply markets, where each buyer has a matching seller. Long and short in the derivatives markets are just opposite sides of the same trade. Consequently, it is a zero-sum game because one trader’s gain is the other’s loss. The short seller in the stock market does something more complex. She borrows shares, sells them, and buys back those shares at a later date to return them to the lender. For futures and forwards, both a long and a short have obligations to transact at a future date. The long agrees to buy the asset, and the short agrees to sell. Options give the buyer the right to buy (call) or sell (put) at a fixed price until a fixed date. No matter whether it’s a call or a put, the buyer who is long has an option. They decide to exercise or not. In contrast, the writer, who holds a short position, has an obligation. The short needs to fulfill the option’s terms as chosen by the long. For this situation, the buyer pays the writer a premium at the start.

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5.7

Order Placement Strategies

All traders face the same dilemma: to trade immediately or to wait to see if one can get a better price. If you want to transact immediately, submit a market order; however, if you can wait for a better price, then submit contingent orders that depend on the price, time, and other relevant factors. But there is a trade-off: market orders may get traded at a bad price, whereas price-contingent orders may never get filled. Example 5.8 considers different types of order placement strategies for trading stocks in fairy-tale (hypothetical) situations. These same order placement strategies also apply to futures and options. Let’s start our discussion with stocks for easy illustration and then show how the argument extends to derivatives like options.

EXAMPLE 5.8: Order Placement Strategies

A Market Order ■

A dealer quotes an ask–offer price of $100.10 and a bid price of $100.00 per share for YBM stock. Hoping that it will rally, you want to buy YBM in a hurry. You can place a market buy order, which gets executed immediately at the best price your broker can find—very likely at $100.10. You may be forced to pay a bit more or enjoy paying slightly less if a spate of buy or sell orders suddenly comes in. Unless there is a severe market breakdown, market orders are always filled.

■

A market order is also the simplest way to buy options and futures. Recall in the previous example that Shorty placed a market sell order and sold the option for $4, which was the bid price. If gold futures prices are quoted at $1,500.10 to sell and $1,500.00 to buy, then a market buy order is likely to get executed at $1,500.10.

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A Price-Contingent Order ■

Sometimes you may get a better price by waiting. You can place a limit order that must be executed at the stated or a better price or not executed at all. One must specify whether this is a day order or a good-until-canceled order. An unfilled day order automatically cancels at the day’s end, while a good-until-canceled (GTC) order stays open almost indefinitely in the specialist’s order book until it is closed. To eliminate stale orders, many brokers automatically cancel GTC orders after a significant time period has passed, for example, 60 days.

■

Suppose you place a limit buy order at $99.00. If the stock price decreases from $100.10 and your trade gets executed, then you obtained a $1.10 price improvement. However, you also run the risk that the stock may rally and the order may never be filled. Similarly for an option on YBM that is being quoted at a bid price of $4.00 and an ask price of $4.05, a trader hoping to benefit from market fluctuations may place a limit sell order at $4.50.

A Time-Contingent Order ■

A time-of-day order specifies execution at a particular time or interval of time. For example, you may request execution at 2:00 pm in the afternoon, or you may request execution between 2:30pm and 3:00pm.

Combinations of Related Orders

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■

Sometimes traders place orders in combination. For example, a spread order may instruct the broker to execute the order when the price difference between two securities reaches a certain value. A cancel order or a straight cancel order completely cancels a previous order. A cancel replace order replaces an existing order with a new one.

Our discussion also applies to options and futures trading, and we can similarly place such orders for bonds, the Treasury securities introduced in Chapter 2.

5.8

Summary

1. A call option gives the buyer (the “long”) the right but not the obligation to buy a specified quantity of a financial or real asset from the seller (the “short”) on or before a fixed future expiration date by paying an exercise price agreed today. A put gives the holder the right to sell. A put is similar to an insurance contract that pays in case of a decline in an asset’s value. The writer is paid a premium for selling such rights. 2. A European option can only be exercised at the maturity date, whereas an American option can be exercised at any time up to and including the expiration date. The names “American” and “European” do not depend on where the option trades. Most exchange-traded options in the US are of the American type. 3. A call holder’s payoff at expiration or if exercised early, its intrinsic value, is the stock price minus the strike price when the stock price is greater than the strike (call is in-the-money) and zero otherwise (call is out-of-the-money). A put’s

QUESTIONS AND PROBLEMS

intrinsic–exercise value is the strike price minus the stock price when the stock price is less than the strike and zero when the stock price is greater than or equal to the strike. Intrinsic value is only part of an option’s value. The rest is the time value, which takes into account that the option’s value may increase before it expires. 4. Payoffs to different option positions as a function of the stock price at expiration generate well-known patterns, which can be depicted in a payoff diagram. Profit diagrams adjust these diagrams for the option’s premium. 5. Traders in a hurry place market orders that are immediately executed. Traders who can wait in the hope of a better price strategically submit limit orders that may be contingent on price, time, and other relevant factors or submitted in combination with other orders. There is a trade-off: market orders may get traded at a bad price, while price improvement orders may never be filled. Contingent orders are possible for trading stocks, options, and futures.

5.9

Cases

Goldman, Sachs, and Co.: Nikkei Put Warrants—1989 (Harvard Business School

Case 292113-PDF-ENG). The case illustrates how investment banks can design, produce (hedge), and price put warrants on the Nikkei Stock Average. Nextel Partners: Put Option (Harvard Business School Case 207128-PDF-ENG).

The case examines issues surrounding Nextel’s shareholders’ vote to exercise a put option that requires the company’s largest shareholder, Sprint Nextel Corp., to purchase all the shares it does not already own. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Pixonix Inc.—Addressing Currency Exposure (Richard Ivey School of Business

Foundation Case 908N13-PDF-ENG. Harvard Business Publishing). The case discusses the impact of exchange rate fluctuations on a firm’s cash flows and how derivatives can be used to manage this risk.

5.10

Questions and Problems

5.1. Explain the differences between an option’s delivery and expiration dates. 5.2. Explain the differences between European and American options. How do the

names relate to the geographical continents? 5.3. Explain the differences between call and put options. Are they just the exact

opposites of each other? 5.4. Explain the differences between long and short in the stock market. Explain

how one shorts a stock. 5.5. Explain the differences between long and short in the option market. What is

the difference between an option buyer and writer? 5.6. Explain the differences between an option’s intrinsic and time values. How do

these relate to the option’s price?

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5.7. Explain the differences between an option’s maturity and exercise date. When

does exercise take place for a European option? For an American option? 5.8. Explain and carefully describe the following four security positions, drawing

payoff diagrams wherever necessary to support your answer: a. short a forward contract with a delivery price of $100 b. short selling a stock at $100 c. going short on an option with a strike price of $100 5.9. Suppose that the current price of platinum is $400 per ounce. Suppose you

expect that in three months the price will increase to $425. You are worried, however, that there is a small chance that platinum may fall below $390 or even lower. What securities can you use to speculate on the price of platinum? 5.10. Why is an American option worth at least as much an otherwise similar

European option? Is there an exercise strategy that one can use to turn an American option into a European option? Explain. 5.11. You ask your broker for a price quote for (a fictional company) Sunstar Inc.’s

March 110 calls. She replies that these options are trading at $7.00. If you want to buy three contracts, what would be your total cost, including commission? Assume that the broker charges a commission equal to a flat fee of $17 plus $2 per contract. 5.12. “Because a call is the right to buy and a put is the right to sell, a long call

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position will be canceled out by a long put position.” Do you agree with this statement? Explain your answer. 5.13. The short position in a put option has the obligation to buy if the option is

exercised. Isn’t this counter to the notion that a short position indicates a “sell”? Explain your answer. 5.14. The following option prices are given for Sunstar Inc., whose stock price equals

$50.00: Strike Price

Call Price

Put Price

45

5.50

1.00

50

1.50

1.50

55

1.00

5.50

Compute intrinsic values for each of these options and identify whether they are in-the-money, at-the-money, or out-of-the-money. 5.15. Do you agree with the following statement: “An out-of-the-money option has

an intrinsic value of zero and vice versa”? Explain your answer.

QUESTIONS AND PROBLEMS

5.16. Compute the profit or loss on the maturity date for a short forward position

with a forward price of $303 and a spot price at maturity of $297. 5.17. Compute the profit or loss on the maturity date for a December 45 call for

which the buyer paid a premium of $3 and a spot price at maturity of $47. 5.18. Compute the profit or loss on the maturity date for a November 100 put for

which the seller received a premium of $7 and a spot price at maturity of $96. 5.19. Because options are zero-sum games, the writer’s payoffs are just the negatives

of the sellers. Demonstrate the following (by algebraic arguments or by using numbers): a. the call seller’s payoff at expiration is –max(0, S – K) = min(0, K – S) b. the put seller’s payoff is –max(0, K – S) = min(0, S – K) 5.20. What is the difference between a market buy order and a limit order? When

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would you use a market versus a limit order?

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6 Arbitrage and Trading 6.1 Introduction

Spread Trading

6.2 The Concept of Arbitrage

Index Arbitrage

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6.3 The No-Arbitrage Principle for Derivative Pricing The Law of One Price Nothing Comes from Nothing

EXTENSION 6.1 Stock Indexes EXTENSION 6.2 Index Arbitrage

6.6 Illegal Arbitrage Opportunities

6.4 Efficient Markets

6.7 Summary

6.5 In Pursuit of Arbitrage Opportunities

6.8 Cases

The Closed-End Fund Puzzle

6.9 Questions and Problems

THE CONCEPT OF ARBITRAGE

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6.1

Introduction

Once upon a time in days not too long ago, a venerable finance professor went with two of his graduate students to a McDonald’s restaurant for a working lunch. One student noticed that the Big Mac sandwiches were on a buy-one-get-one-free special. Bubbling with excitement, the student ran to the mentor to share the news. His jaws dropped as he approached, for the master himself was sitting with two Big Macs, discussing the selling of one to the second student at half price: “You also know about this?” “Yes,” the professor replied, “it’s an arbitrage.” Buying two for the price of one and selling the second free sandwich at half price would indeed be an arbitrage opportunity! Taking a cue from this embellished story, we define arbitrage (arb) as a chance to make riskless profits with no investment. The concept of arbitrage is invaluable for building pricing models, which are mathematical formulas that price derivatives in terms of related securities. Since the 1970s, financial markets have been generating innovative derivatives by attaching extra features to simple financial securities, by designing entirely new derivatives with complicated payoffs, or by combining several derivatives into a single composite security. For this last example, the composite security’s price must equal the sum of the prices of these simpler constituent derivatives. Otherwise, eagle-eyed traders can make arbitrage profits by buying the relatively cheaper of the two and selling the more expensive one. This observation leads one to recognize that a more general truth holds. In the absence of arbitrage, all portfolios or securities with identical future payoffs must have identical values today. This is the essence of the no-arbitrage principle that we discuss later in this chapter. Easy arbitrage opportunities are hard to find. Still we start with them because they form the foundations of our study and help us understand the more complex arbitrages that appear in today’s marketplace. The existence of arbitrage opportunities implies that the markets are somehow inefficient. We introduce the various notions of market efficiency and discuss what happens when they are violated. Next we talk about some pricing anomalies and the trading strategies that profit from them.

6.2

The Concept of Arbitrage

Financial markets exert a ruthless discipline on their players. No one advises a careless trader to modify bad quotes; instead, people grab such mispricings, and the misquoting trader sees his wealth disappear. Earlier we defined arbitrage as a chance to make riskless profits with no investments. This trivial description masks an activity that makes and takes millions of dollars every day. Eager traders pour over computer screens displaying prices from diverse markets and scan them for arbitrage opportunities. The act of arbitrage supports the livelihood of many. But gone are the days when one could sit in an armchair reading the financial press and find easy arbitrages. Today’s arbitrageurs work very hard for their money.

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Traders searching for arbitrage is rational and predictable behavior. Picking up a hundred dollar bill from the ground is normal behavior.1 It is hard, if not impossible, to model behavior if one oftentimes picks up the bill, sometimes doesn’t, and yet at other times picks up the bill but leaves a few bucks in its place for the person coming next! Therefore, for the subsequent theory, we assume that one never leaves money behind and relentlessly picks up even the tiniest of arbitrage opportunities. As we later see, this assumption has very powerful implications; indeed, it lies at the heart of the derivative pricing models presented in this book. There are many types of arbitrage. Some arbitrage operations are easy to understand and implement but hard to find in practice, whereas others are just the opposite. All such arbitrages can be decomposed into one of two types involving mispricings across space and mispricings across time. An example of mispricing across space is when the sum of the parts’ prices differ from the price of the whole. Example 6.1 discusses these two types of arbitrage opportunities.

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EXAMPLE 6.1: The Two Types of Arbitrage Opportunities ■

Arbitrage across space (see Figure 6.1). Suppose Your Beloved Machines Inc.’s (YBM) stock has an ask price of $100.00 on the Big Board and a bid price of $101.00 in Tokyo. If the brokerage cost is $0.10 per share, then one can buy stocks at a lower price in New York, simultaneously sell them at a higher price in Tokyo, and make 80 cents in arbitrage profits.

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Example (arbitrage across space). The sum of the parts’ prices differs from the price of the whole. Some people see arbitrage opportunities in conglomerates, which are huge companies with diverse lines of business. They claim that the share prices of conglomerates often fail to reflect the true value of the underlying assets and trade too low. Corporate raiders buy conglomerates, sell the constituent businesses or their assets, and more than recover their initial investments. “I saw the company and thought that the sum of the parts was worth more than what it was trading for,” observed a notorious corporate raider Asher Edelman as he targeted a French conglomerate.2 Investment bankers doing mergers and acquisitions believe the opposite. They believe that mergers create arbitrage opportunities where the sum of the parts is less than the whole. The idea that the same payoffs no matter how they are created, trade at the same price is known as the law of one price.

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Arbitrage across time (see Figure 6.1). Free lottery tickets are an example of arbitrage across time. Result 3.1 of Chapter 3, which states that a stock’s price falls exactly by the amount of the dividend on the ex-dividend date, if violated, also represents an example of an arbitrage across time.

1

In an often-told tale, two finance professors are walking together, and they see a $100 bill lying on the ground. When one ponders whether he should pick it up, the other replies not to bother because the bill is not really there: “As markets are efficient, someone must have already picked it up before.” This story is usually told to clarify the “ideal” notion of an efficient market, while teasing finance academics.

THE CONCEPT OF ARBITRAGE

FIGURE 6.1: The Two Types of Arbitrage Opportunities Security trades at some price Buy cheap, sell expensive; make net profit after paying brokerage costs Same security trades elsewhere at a lower price

i)

ii)

Now

Later

Zero net investment

Nonnegative payoff always and strictly positive payoff with positive probability

Positive cash flows

Nonnegative payoff always

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2

Corporate raider Asher B. Edelman once taught a business course at Columbia University, in which he offered $100,000 to any student who would find a takeover target for him. However, the university banned the offer. In the late 1990s, he targeted Societe du Louvre, the French holding company to which the above comment applies (see “Champagne Taste: An American Raider in Paris Challenges an Old World Dynasty—Edelman Presses Taittingers to Choose Family Values or Shareholder Value—Free to Paint a Hotel Pink,” Wall Street Journal, November 11, 1998).

Are such arbitrage opportunities likely to exist in today’s financial market, where technological advances have lowered brokerage costs and made it possible to move millions of dollars in a matter of seconds? Most economists believe that the answer to this question is no—and if they exist, it is especially hard to find them in wellfunctioning and efficient markets. Perhaps they can be found in some emerging markets, but they fade away quickly as traders start grabbing them. Sometimes what may seem like arbitrage opportunities in a perfect world (where market imperfections or frictions like transaction costs, bid/ask prices, taxes, and restrictions on asset divisibility are assumed away) cease to be so when market imperfections are included. In actual markets, market frictions are relevant and should always be taken into account when considering arbitrage opportunities. Let’s discuss a few real-life examples of arbitrages.

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EXAMPLE 6.2: Real-Life Arbitrages ■

Sports arbitrage. Not too long ago the soccer World Cup finals had two powerhouse teams, Brazil and Germany, playing. Uncle Sanjoy relishes the story of how he visited a Wall Street firm and entered into two bets with acquaintances. First, he agreed to pay one person $10,000 if Brazil won but receive $5,000 otherwise. Then, uncle slyly adds that he visited another office, and bet with a second person to receive $11,000 if Brazil won but pay $4,000 otherwise. No matter which team won, uncle locked in $1,000. Although gambling exasperates his wife, uncle quietly commented that he had locked in arbitrage profits by exploiting inefficiencies in the system and that he truly enjoyed watching the game!

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Airline miles arbitrage. Airlines put all kinds of restrictions on tickets, serve scant morsels of food, manhandle baggage, make passengers endure long delays, and cancel flights. Those who have suffered in the sky at the hands of airlines will be amused by a story in the Wall Street Journal, “How Savvy Fliers Make the Most of Their Miles” (December 16, 2008), which related how some travelers got the best out of their frequent flier miles. Normally a ticket obtained by redeeming frequent flier miles would yield the value of approximately one cent per mile. Some folks figured out how to transform this once cent per mile into more. One gentleman paid about 1.3 cents per mile to “friends” through the Internet and then used the miles for business class tickets between China and the United States. “I call it airline miles arbitrage,” said the savvy traveler, who created business class tickets for coach ticket prices and in the process, earned about six to nine cents per mile.

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Uncle Sam’s generosity. The idea of arbitraging Uncle Sam (the US government) appeals to most people, but few succeed in doing so. However, an article in the Wall Street Journal, “Miles for Nothing: How the Government Helped Frequent Fliers Make a Mint—Free Shipping of Coins, Put on Credit Cards, Funds Trip to Tahiti; ‘Mr. Pickles’ Cleans Up” (December 7, 2009), reported a story where this happened. What happened? Because a paper dollar lasts only about twenty-one months in circulation, dollar coins save a country money as they can last for thirty years or more. To promote their use, the US Mint offered to sell presidential and Native American $1 coins at face value, ship them for free, and buyers could charge the purchase to their credit cards without handling fees. However, wellintentioned plans sometimes have unintended consequences. Arbitrageurs bought large quantities of coins from the mint, paid for them with credit cards that offered frequent flier miles for free air travel (typically, one cent for each dollar charged in purchases), and immediately deposited the coins in the bank to pay for the credit card coin purchase. The story states that one software consultant ordered $15,000 worth of coins and even had the delivery person deposit the coins directly in his car’s trunk so that he could take them to the bank. Discovering such activities, US Mint officials sent letters to prospective buyers seeking the reasons for their purchase and denied further access to those who did not respond.

THE NO-ARBITRAGE PRINCIPLE FOR DERIVATIVE PRICING

6.3

The No-Arbitrage Principle for Derivative Pricing

Modern finance theories use the idea that arbitrage opportunities do not exist in well-functioning markets to derive many important results. This was pioneered by the Miller and Modigliani (M&M) propositions in corporate finance, which state that if market imperfections like taxes are assumed away, then the choice of debt or dividend policy does not affect the value of a firm. (Taxes are imperfections! Who says finance jargon is boring?) In our case, the absence of arbitrage opportunities lies at the heart of derivative pricing models, including the celebrated Black–Scholes– Merton and the Heath–Jarrow–Morton models included in this text. This section explains how.

The Law of One Price

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Suppose one wants to price a market-traded derivative, which we label as “portfolio A.” Assume that there are no cash flows on this traded derivative before maturity so that there are only two dates of concern—today and when the derivative matures. Now let us create portfolio B, a collection of different traded securities with known prices and no intermediate cash flows, to match portfolio A’s payoffs on the maturity date (see Figure 6.2). The values of these two portfolios are linked by a no arbitrage principle: to prevent arbitrage, two portfolios with identical future payoffs must have the same value today.3 Using this principle, the cost of creating portfolio B is the “fair” or “arbitrage-free” value of the traded derivative portfolio A.4 This is known as the law of one price.

Nothing Comes from Nothing Continuing with the previous discussion, if we subtract portfolio A from portfolio B, then we get the second representation, which will be one of our main methods for proving results (see Figure 6.2). By construction, the value of this new portfolio is zero for sure on the maturity date. To prevent arbitrage profits, it must also have zero value today. The derivative’s price can be solved from this condition. This insight can be paraphrased as nothing comes from nothing.5 When identical cash flow securities are subtracted, yielding zero at a future date, the result is emptiness, nothing, nada, zero now! These are the two standard techniques used for pricing derivatives from Chapter 11 onward in this book. 3

Suppose this principle does not hold. If portfolio A is cheaper than portfolio B today, “buy low, sell high”: buy A and sell B. You will have a zero cash flow in the future because the cash inflow will cancel the outflow, and the price difference you capture today becomes arbitrage profit. If B is cheaper than A, just reverse the trades to capture the arbitrage profit. 4

The same argument works when the price of the derivative is known but there is an unknown variable in the payoff. For example, a forward contract has a zero value on the starting date, but the unknown forward price is in the payoff. The solution to this equation is the arbitrage-free forward price. 5 “Nothing comes from nothing” (or ex nihilo nihil fit in Latin) is a philosophical expression of a thesis first advanced by the Greek philosopher Parmenides of Elea, who lived in the fifth century bc.

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FIGURE 6.2: Pricing Derivatives The law of one price Today

Later date

Portfolio A

Must be equal; No intermediate cash flows otherwise arbitrage

Portfolios have the same final value

Portfolio B

Nothing comes from nothing Now

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Portfolio A – (Portfolio B)

Later date No intermediate cash flows

Zero value (for sure)

Must be zero; otherwise arbitrage

As we have often observed, arbitrage opportunities are likely to be absent in wellfunctioning and efficient markets—but what do we really mean by an “efficient market”? Well, read on.

6.4

Efficient Markets

When a finance academic states that “markets are efficient,” she means an informationally efficient market: a market where asset prices quickly absorb and “fully” reflect all relevant information. But what relevant information is this referring to? The efficient markets hypothesis (EMH) is presented with respect to three different information sets: 1. Weak-form efficiency asserts that stock prices reflect all relevant information that can be gathered by examining current and past prices. If this is true, then technical analysis (which involves looking at price patterns to predict whether a stock’s price will go up or down) will not generate returns in excess of the

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

risk involved. In addition, and more relevant to us, if the market is weak-form efficient, then there are no arbitrage opportunities. Why? Because finding arbitrage opportunities only depends on the information contained in current and past prices. It is this connection that relates an efficient market to the arbitrage-free pricing of derivatives. 2. Semistrong-form efficiency asserts that stock prices reflect not only historical price information but also all publicly available information relevant to those particular stocks. If this holds, then fundamental analysts (fundamental analysis involves reading accounting and financial information about a company to determine whether a share price is overvalued or undervalued) would be out of business.

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3. Strong-form efficiency asserts that stock prices reflect all relevant information, both private and public, that may be known to any market participant. If this is true, then even insiders (who have privileged, yet to be made public information about the company) will not make trading profits in excess of the risk involved. There is a never-ending debate on whether markets are efficient with respect to these three different information sets. The preponderance of evidence tends to accept weak-form efficiency, and strongly reject strong-form efficiency, especially given the numerous insider-trading lawsuit convictions historically obtained by the Securities and Exchange Commission (SEC). The evidence with respect to semistrong-form efficiency is mixed. The assumption of no arbitrage underlying the models developed in this book is very robust. Indeed, if markets are weak-form efficient, then no arbitrage opportunities exist. Hence the weakest form of market efficiency justifies the assumption of no arbitrage. But don’t get confused. Assuming there are no arbitrage opportunities in subsequent modeling does not imply that we are assuming that the market is weak-form efficient. As just discussed, the assumption of no arbitrage is a much weaker notion than even a weak-form efficient market. Hence it is more likely to be satisfied in actual markets, and thus it provides a more robust model than market efficiency does.

6.5

In Pursuit of Arbitrage Opportunities

Well-resourced, astute players trade for the purpose of exploiting arbitrage opportunities. As the next section illustrates, academics have uncovered phenomena that seem to suggest the existence of such arbitrages.

The Closed-End Fund Puzzle Closed-end fund share prices are often claimed to represent arbitrage opportunities. To understand this claim, we must first understand closed-end funds, and to understand closed-end funds, we need to first understand investment companies. An investment company sells shares to investors, invests the funds received in various securities according to a stated objective, and passes the profits and losses back to the investors after deducting expenses and fees. Federal securities law classifies investment companies into three basic types: mutual funds (legally known as open-end

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companies), closed-end funds (legally known as closed-end companies), and UITs (legally known as unit investment trusts). Mutual funds, like those offered by Fidelity and Vanguard, are open ended and take new investments at any time. A shareholder can redeem his shares directly from the fund whenever he desires. Closed-end funds issue shares at their creation but none thereafter. Just like regular stocks, the shares of closed-end funds trade on organized exchanges. Closed-end funds are not redeemed by the fund for cash. To close out an investment, an investor must sell her shares on the secondary market. Finally, unit investment trust shares can be redeemed like mutual funds; moreover, they have a termination date when the fund’s assets are sold and the proceeds are paid out to investors. Redemptions in mutual funds and UITs pay the company’s net asset value per share. Net asset value (NAV) is the value of the company’s total assets minus its total liabilities, often expressed on a per share basis. For example, if an investment company has securities and other assets worth $110 million and has liabilities of $5 million, its NAV would be $105 million. If the investment company has issued 1 million shares, then $105 would be its NAV per share. NAV changes daily due to market fluctuations. Mutual funds and UITs are usually required to compute their NAV at least once per business day (which they usually do after the market closes). Closed-end funds, however, are exempted from this requirement. For reasons that are doubted and debated, the shares in most closed-end funds trade at a discount from their asset value. This is known as the closed-end fund puzzle. These discounts represent potential arbitrage opportunities. To see this, suppose that a closed end fund is worth $100 million in the market, but the underlying securities have a net asset value of $105 million. What can one do? One can buy all the shares, open up the fund, and sell the underlying securities to make $5 million in arbitrage profits. However, not all closed-end funds trade at a discount. Some trade at premiums over their underlying asset values, like the legendary Warren Buffett’s Berkshire Hathaway Company at the end of the twentieth century. Why? Again, a puzzle.

Spread Trading At the heart of many arbitrage strategies is spread trading. Spread trading involves buying and selling one or more similar securities when their price differences are large, with the aim of generating arbitrage profits as their price differences later converge to some target value. For example, suppose that Your Beloved Machines Inc. (YBM) has two classes of shares with different voting rights. You find that class A shares are trading at $100 and class B shares are trading at $85. This is a spread of 100 – 85 = $15. However, you’ve discovered that the spread has historically been $10, and you expect it to return to that level. To take advantage of this price discrepancy, you can buy the underpriced class B shares and sell the overpriced class A shares. If your bet is correct and the spread returns to $10, you can reverse the trades and capture $5 in profits. Notice that you are betting on the movement of a spread and not on the direction of the stock prices

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

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themselves. Spread trading is one of the most common arbitrage trading strategies used in financial markets. The next section discusses one such widely used strategy, called index arbitrage.

Index Arbitrage A stock index is an average of stock prices that are selected by some predetermined criteria. For example, the Dow Jones Industrial Average (also called the DJIA, Dow 30, the Dow Industrials, the Dow Jones, or simply the Dow) and the Standard and Poor’s 500 Index (S&P 500) are stock indexes created by using a price-weighted average of thirty major US stocks and the market value–weighted average of five hundred major US companies, respectively (see Extension 6.1 for a discussion of indexes). Sometimes the value of a stock market index gets disjointed from the arbitrage-free price of a futures contract written on the index. In such situations, the potential for index arbitrage exists. Suppose the traded futures contract on an index is overvalued. The arbitrage is to short the traded futures contract and go long a synthetic futures contract constructed to have the same cash flows as the traded futures contract. The synthetic futures contract is constructed using the portfolio of the traded stocks underlying the index. This strategy will lock in the price difference at time 0 and have zero cash flows thereafter. The arbitrage is a violation of the nothing comes from nothing principle. Extension 6.2 illustrates index arbitrage via a simple example.

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EXTENSION 6.1: Stock Indexes There are two basic types of stock indexes: price-weighted and value-weighted indexes. We illustrate both using examples.

EXT. 6.1 EX. 1: A Price-Weighted Index (PWI) ■

Holding one share of each stock, summing the share prices and dividing by the total number of shares creates a price-weighted index. Consider the hypothetical stock price data in Ext. 6.1, Table 1. Here YBM’s share price is $100 today and $110 tomorrow. BUG’s price is $50 today and $45 tomorrow. The equal average PWI is equal to Sum of stock prices / Number of stocks = (100 + 50) /2 = 75 today and (110 + 45) /2 = 77.5 tomorrow

■

While one stock went up and the second went down, the PWI index registered a 2.5 point gain or (Tomorrow’s index value – Today’s index value) / Today’s index value = (77.5 – 75) / 75 = 0.0333 or a 3.33 percent increase

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This is called a price-weighted average because when viewed as a portfolio, each stock’s return is weighted by a percentage proportionate to the price of the stock relative to the value of the portfolio.

The DJIA is computed by creating a price-weighted average of thirty high-quality blue chip stocks from diverse industries. The component stocks are selected by the editors of the Wall Street Journal. The membership list is periodically modified so that the index continues to accurately reflect the general US stock market. Such revisions are imperative because companies go through mergers, acquisitions, and bankruptcy.

EXT. 6.1 TABLE 6.1: Stock Price Data Stock

Today’s Price

Tomorrow’s Price

Shares Outstanding (millions)

Today’s Market Value (millions)

Tomorrow’s Market Value (millions)

YBM

$100

$110

50

$5,000

$5,500

BUG

$50

$45

20

$1,000

$900

$6,000

$6,400

Total

EXT. 6.1 EX. 2: A Value-Weighted Index (VWI)

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■

Multiplying the stock price by the shares outstanding (which gives the stock’s total market capitalization) and then summing across all stocks creates a value-weighted index. Using the data from Ext. 6.1 Tab. 1, we get a VWI equal to Sum of stock price times shares = (100 × 50 + 50 × 20) = 6, 000 today and (110 × 50 + 45 × 20) = 6, 400 tomorrow The weights (percentage holdings of each stock) in the index are for YBM = $5, 000/$6, 000 = 0.8333 for BUG = $1, 000/$6, 000 = 0.1667

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The percentage change in the VWI is (Tomorrow’s index value – Today’s index value) /Today’s index value = (6, 400 – 6, 000) /6, 000 = 0.0667 or a 6.67 percent increase

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In practice, the index is often normalized to make its value equal to some arbitrary number at a given date. This is for convenience in reporting the index. For example, the VWI index’s value could be normalized to be 100 today by dividing by 60. The number 60 must also divide the index’s value at all future dates. In this case, the index value tomorrow would be 6,400/60 = 106.67.

IN PURSUIT OF ARBITRAGE OPPORTUNITIES

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This is called a value-weighted index because when viewed as a portfolio, each stock’s return is weighted by its market value relative to the total value of the stock market. The S&P 500 is a market-value-weighted index of five hundred large-sized US companies. A value-weighted index is sometimes called a market-cap weighted index.

EXTENSION 6.2: Index Arbitrage This extension explains index arbitrage. To simplify the presentation, the discussion replaces futures with forward contracts. For the purposes of this demonstration, this difference is not important. The differences between forward and futures contracts, which were discussed in Chapter 4, will be further clarified in subsequent chapters. Index arbitrage generates riskless profits by putting the no-arbitrage principle to work. We illustrate it with a simple example.

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EXT. 6.2 EX. 1: The No-Arbitrage Principle ■

Let a PWI be created by computing an average of the stock prices of YBM and BUG worth $100 and $50 today, respectively. This gives an index value of $75 today. - Suppose that a dealer quotes a forward price of $80 on a forward contract on PWI that matures in one year. - Let the simple interest rate be 6 percent per year. And suppose that neither stock will pay any dividends over the next year.

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Next, suppose that a trader believes the forward price is too high. Intuitively, she wants to sell the traded forward contract and create a synthetic forward contract by buying the PWI index portfolio, borrowing the funds to do so, and holding the portfolio until the traded forward contract matures. She performs the following steps: - She sells two forward contracts to the dealer, whereby she locks in a selling price of $80 per unit of the index: $160 in all. - Simultaneously, she buys one share of each stock for a total cost of $150 and finances this purchase by borrowing. Notice that she has a zero cash flow today because the traded forward requires no cash flows on its initiation date. - One year later, she gives two units of the index to the dealer who sold her the forward. This costs nothing because she made allowance for this by purchasing the stocks earlier. She gets $160 from him in return. - But she has to repay the loan with interest. This entails an outflow that equals Today’s index purchase price × (1 + interest cost) = 150 × 1.06 = $159 - She pays out $159 but receives $160, and makes $1 in arbitrage profits. She makes this profit no matter where the index moves a year from today. We make extensive use of this technique to develop cost-of-carry models in Chapters 11 and 12.

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6.6

Illegal Arbitrage Opportunities

Arbitrage is a broad concept. You can bring under its umbrella dubious, socially unfavorable, and illegal trading activities. Medical students first learn how a healthy body functions before they study abnormalities and diseases. We follow a similar approach. Previously we introduced legal arbitrage opportunities that facilitate market efficiency. Now we shift gears and study nefarious arbitrage opportunities. These illegal arbitrage opportunities occur in malfunctioning securities markets and remain a major problem that regularly appears in the national, local, and financial press. Trading abuses and manipulations are forms of illegal arbitrage opportunities that we now discuss. One such abuse is front running, which is trading based on an impending transaction by another person. For example, a broker may buy on his own account in front of his customer’s buy order. Then, he makes unscrupulous gains by selling the shares he bought to the customer at a higher price when executing the customer’s buy order. A chronic problem facing free markets is that individuals often try to manipulate prices to their advantage. Manipulation has long been a concern in the securities market. For example, in its report on the [Securities] Exchange Act of 1934, the Senate Committee wrote, “The purpose of the act is... to purge the securities exchange of these practices which have prevented them from fulfilling their primary function of furnishing open markets for securities where supply and demand may fully meet at prices uninfluenced by manipulation or control.”6 The Commodity Exchange Act states that it is illegal to manipulate or attempt to manipulate a commodity’s future delivery price, yet proving manipulation in a court of law is not easy. Next we discuss some well-known examples of market manipulation. A bear raid is a market manipulation strategy generating arbitrage profits that was widely prevalent in US stock markets in the nineteenth and early twentieth centuries. Remember from Chapter 2 how short selling works—a short seller borrows shares, accepts the obligation to pay dividends to the lender, and agrees to return the security to the lender at a future date. Traders organizing a bear raid take large short positions in a company’s stock, spread unfavorable rumors that depress the stock further, and buy back those shares when other shareholders panic and unload their stock holdings at depressed prices. Short sellers can get severely burned if they get caught in a market corner and a short squeeze, which was also a pervasive problem in the early days of stock markets. The next example shows how a market corner and short squeeze works.

6

Report of the Committee on Banking and Currency, “Stock Exchange Practices,” U.S. Senate Report No. 1455, 73d Cong., 2d Sess., 1934 (the “Fletcher Report”). The quote appears in Chapter I “Securities Exchange Practices,” Section 11 “The government of the exchanges,” Subsection (b) “Necessity for regulation under the Securities Exchange Act of 1934.”

ILLEGAL ARBITRAGE OPPORTUNITIES

EXAMPLE 6.3: A Market Corner and Short Squeeze ■

Suppose a company has ten thousand outstanding shares, which sell for $50 in the market. This is the deliverable supply, of which Alexander owns three thousand shares, Bruce owns three thousand shares, and Caesar owns four thousand shares. The shares are held in street name with their common broker, Mr. Brokerman, so that short sellers can borrow (see Figure 6.3).

FIGURE 6.3: Example of a Market Corner and Short Squeeze

Shorty borrows shares from Alexander

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Alexander 3,000 shares

Caesar 4,000 shares

Bruce 3,000 shares

Shorty Short 3,000 shares

Sells shares to Ruth Ruth owns 13,000 shares Short sells to Ruth

Ruth controls 13,000 shares, though the deliverable supply is 10,000 shares. Ruth has cornered the market. She will squeeze Shorty by charging him a high price for 3,000 shares that he has to buy and return to Alexander.

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Shorty is bearish on the company and expects the share price to decline from $50 to $40. He short sells three thousand shares, which his broker, Mr. Brokerman, borrows from Alexander. They are sold at $50 per share to Ruth, who happens to be a “ruthless manipulator.”

■

Ruth now buys all the shares from Alexander, Bruce, and Caesar. She now has a monopoly position and has bought three thousand more shares than the deliverable supply of ten thousand shares, making it thirteen thousand in all. Ruth has cornered the market.

■

At the time of settlement, Alexander does not have shares to give to Ruth. He asks Mr. Brokerman to return his three thousand shares. Since Ruth owns them all, nobody is willing to lend shares. Normally, Shorty can keep his short position open as long as he likes. As scarcity is preventing Alexander’s trade from getting settled, Brokerman informs Shorty that he has to return the borrowed shares at once.

■

Shorty has nowhere to go but Ruth to buy shares. This is a case of congestion: Shorty, trying to cover his short position, finds that there is an inadequate supply of shares to borrow, and Ruth is unwilling to sell her shares, except at sharply higher prices. In other words, Ruth is squeezing the short. She can easily bankrupt Shorty.

This example easily extends to futures market. First the trader goes long the futures contract in excess of the immediately deliverable supply. Then she keeps her long position open, eventually acquires all the deliverable supply, and ends up in a monopoly position. After she has cornered the market, a short squeeze develops. She demands delivery, but the short cannot find any supply to cover their positions, except from her—and she extracts her price. These undesirable situations are in violation of the Commodity Exchange Act statute. To stop short squeezes, the CFTC can suspend trading and force settlement at a “fair” price set by the exchange issuing the contract. This happened, for example, with the Hunt brothers’ holdings of silver futures in 1979 and 1980. Contract provisions are also designed to increase the deliverable supply, which minimizes the likelihood of manipulation, by allowing variation in the quality of the asset delivered (with appropriate price adjustments) and a longer delivery period. Still, manipulations happen. We discuss additional futures market manipulations in Chapter 10.

6.7

Summary

1. Arbitrage is a chance to make riskless profits with no investment. Arbitrage profits can be generated in several ways—across time and across space. An example of an arbitrage across space is when the sum of the parts’ prices differ from the price of the whole. 2. The no-arbitrage pricing principle involves skillfully creating a portfolio of assets that replicates the payoff to a traded derivative and then arguing that the cost of this portfolio is the “fair” or “arbitrage-free” value of the traded derivative. This is the standard technique used for pricing derivatives.

QUESTIONS AND PROBLEMS

3. Arbitrage profits do not exist in an efficient market where market prices reflect all past and current price information. However, markets aren’t always efficient, and sophisticated traders use advanced tools and techniques to make arbitrage profits. 4. A chronic difficulty with free markets is that traders try to manipulate prices to their advantage. Market manipulation is an example of an illegal arbitrage opportunity. 5. A market corner and short squeeze is a common manipulation technique that has a checkered past in futures markets. It involves a trader going long in the futures market and squeezing the shorts subsequently in the cash market, when they scramble to cover their short positions.

6.8

Cases

Arbitrage in the Government Bond Market? (Harvard Business School Case 29-

3093-PDF-ENG). The case examines a pricing anomaly in the large and liquid Treasury bond market, where the prices of callable Treasury bonds seem to be inconsistent with the prices of noncallable Treasuries and an arbitrage opportunity appears to exist. RJR Nabisco Holdings Capital Corp.—1991 (Harvard Business School Case 29-

2129-PDF-ENG). The case explores a large discrepancy in the prices of two nearly identical bonds issued in conjunction with a major leveraged buyout and considers how to capture arbitrage profits from the temporary anomaly.

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Nikkei 225 Reconstitution (HBS Premier Case Collection Case 207109-PDFENG).

The case considers how an institutional trader who may receive several billion dollars of customer orders in connection with a redefinition of the Nikkei 225 index can provide liquidity for the event and pick up arbitrage profits.

6.9

Questions and Problems

6.1. What is an arbitrage opportunity across space? Give an example. 6.2. What is an arbitrage opportunity across time? Give an example. 6.3. What is the law of one price? 6.4. Explain the pricing principle “nothing comes from nothing.” 6.5. Suppose a two-year Treasury note is trading at its par value $1,000. You

examine the cash flows, and if you sell them individually in the market, you get $47.85 for the six-month coupon, $45.79 for the one-year coupon, $43.81 for the one-and-a-half-year coupon, $41.93 for the two-year coupon, and $838.56 for the principal. a. Are these prices correct? b. If not, show how you can capture arbitrage profit in this case.

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6.6. Suppose a two-year zero-coupon bond has a price of $0.90 and a three-year

zero has a price of $0.85. A bank allows you to borrow or lend at 4 percent, compounded once a year. Show two ways that you can make arbitrage profits from these prices. 6.7. You can trade Boring Unreliable Gadget Inc.’s stock for $77 per share in the

United States and for €50 in Europe. Assume a brokerage commission of $0.10 per share in the United States and €0.10 in Europe. A foreign exchange dealer quotes a bid price of $1.5000 for each euro and offers them for $1.5010. a. Are these prices correct? b. If not, show how you can capture arbitrage profit by trading BUG stock. 6.8. What is an efficient market, and what does it mean for a market to be weak-

form, semistrong-form, and strong-form efficient? 6.9. If the market is weak-form efficient, do arbitrage opportunities exist? Explain

your answer. Do you think arbitrage opportunities exist? Explain your answer. 6.10. What is a closed-end fund, and what is the “closed-end fund puzzle”? 6.11. Suppose that Boring Unreliable Gadget Inc. has two classes of shares with

different voting rights. You find that class A and class B shares are trading at $49 and $37, respectively. However, historically, the spread has been $15, and you expect the price difference to reach that level. a. Explain how you would set up a spread trade and how much profit you

expect to make once the prices correct themselves.

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b. Would the preceding strategy work if class A stock goes up to $75 per share? 6.12. What is a stock index? Describe the differences between the Dow Jones

Industrial Average and the S&P 500 stock price indexes. 6.13. What is algorithmic trading, and how can it be used to make profits? How does

this relate to program trading? 6.14. What is front running? Explain why an investor should be concerned if his

stockbroker front runs his trades. 6.15. What is a short squeeze, and under what circumstances does it occur? 6.16. What is a bear raid? How does a bear raid relate to trading on inside

information?

7

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Financial Engineering and Swaps 7.1 Introduction

Interest Rate Swaps

7.2 The Build and Break Approach

Forex Swaps

7.3 Financial Engineering

Currency Swaps

Cash Flows versus Asset Values

EXTENSION 7.1 Valuing Fixed-for-Fixed Currency Swaps

Examples

Commodity and Equity Swaps

7.4 An Introduction to Swaps 7.5 Applications and Uses of Swaps 7.6 Types of Swaps

Credit Default Swaps

7.7 Summary 7.8 Cases 7.9 Questions and Problems

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7.1

Introduction

In January 1993, the French chemical and pharmaceutical giant Rhône-Poulenc offered its employees an opportunity to buy discounted shares as part of its forthcoming privatization plan. With an eye toward “wide employee participation,” the scheme involved a dizzying array of features, including minimum investment incentives, price discounts, free shares, interest-free loans, tax benefits, an option to opt in or out of dividends, and an installment plan to pay for the shares over time. Despite the best intentions of this intricate scheme, the plan badly flopped. Then came Bankers Trust, an American bank famous for its derivatives and trading prowess (and infamous for widely publicized losses incurred by its clients Procter & Gamble and Gibson Greetings). Bankers Trust streamlined the terms of the Rhône-Poulenc plan and also added a critical safety feature—the investment would earn a guaranteed minimum return but surrender some gains if the stock price increased. The modified plan was a success. How could they guarantee a minimum return? Armed with your derivatives knowledge, you can now understand what happened. The plan protected RhônePoulenc’s employees from the downside (by buying a put option), and it financed this plan by removing some of the upside (by selling a call option). This is an example of financial engineering. Financial engineering studies how firms design derivatives to solve practical problems and exploit economic opportunities. This vast subject is taught at varying depths in many universities around the world. After presenting some simple examples, we introduce swaps, which were among the first uses of financial engineering. We briefly discuss applications and uses of swaps and illustrate the basic workings of interest rate swaps, forex swaps, currency swaps (including a valuation formula), commodity swaps, equity swaps, and the highly useful but much maligned credit default swaps. Chapter 22 discusses swap markets and interest rate swaps again in greater detail.

7.2

The Build and Break Approach

Sometimes children snap together plastic blocks to build beautiful toys. At other times, they rip apart their creations into simpler structures. We adopt a similar build and break approach. Sometimes we combine derivatives and financial securities to create new derivatives tailored to meet investment needs, and at other times, we break down a complex derivative into its simpler components. This is important for two reasons. The first reason is that it provides an approach for pricing derivatives—break a derivative into simpler parts, price the simpler parts, and sum to get the original derivative’s price. Arbitrage, like adhesive, binds the parts’ prices together to equal the price of the whole and makes sure that the technique works. For example, you can price a convertible bond as the sum of a regular bond and an option to convert the bond into a fixed number of stocks. An extendible bond may be valued as the price of

FINANCIAL ENGINEERING

a straight bond plus an option to extend the bond’s life. So when a rocket scientist tries to sell you a complex derivative at an exorbitant price, you can value it by this method and prevent yourself from being gouged. The second reason is that it enables the use of derivatives to modify a portfolio’s return to meet various investment objectives. For example, suppose a company is considering issuing fixed rate debt when interest rates are high to finance an investment project. The company is worried about getting stuck with an expensive debt issue if interest rates go down. To avoid this problem, the company can issue callable bonds. In the case that interest rates decline, the company can exercise the call provision in the callable bonds and buy back the bonds at a predetermined price. Where does the company get the funds to buy back the bonds? It issues new fixed rate bonds at the lower interest rate.

7.3

Financial Engineering

Toward the end of the 20th century, financial engineering emerged as a vibrant, new subfield of finance and operations research.1 Financial engineering applies engineering methods to finance. At the simplest level, financial engineering and risk management puts our build and break approach to work. The best way to understand this approach is through some examples of financial engineering in practice.

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Cash Flows versus Asset Values To a beginning student, when forming and liquidating portfolios, the signs of the cash flows and asset values may be confusing. To preempt such confusion, we make the following observations. When constructing a portfolio, a positive value means that you are purchasing an asset, hence there is a negative cash flow. For example, if you are buying Your Beloved Machine stock, it shows up as a positive value in the portfolio, but you spend $100, a negative cash flow, in acquiring it. If the value is negative, you are creating a liability, and there is a positive cash flow. For example, borrowing $90 gives a positive cash inflow of $90. The $90 is a liability because you must pay it back in the future. Note that when a portfolio is constructed, the signs of the cash flows and asset values are opposite. At liquidation, this relationship changes. When liquidating a positive value position, you are selling an asset; hence there is a positive cash flow. For example, YBM has a positive value in your portfolio, and selling it for $120 will generate a positive cash flow. If the value is negative before liquidation, then the cash flow is negative because you are closing a liability. For example, your $90 loan, being a liability, has a negative value in the portfolio. Repaying it on the liquidation date, say, for $94, incurs a negative cash flow. Note that at liquidation, the signs of the cash

1

Cornell University offered the first financial engineering degrees. It states on the Cornell University website (www.orie.cornell.edu/orie/fineng/index.cfm) that “Robert Jarrow and David Heath advised students for several years before formalizing the program in 1995, making Cornell one of the very first universities to have a graduate program in Financial Engineering, and arguably the oldest such program in the world.”

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flows and asset values are the same. We will often call both the cash flows and values at liquidation the payoffs. As they have the same sign, no confusion should result. When proving results, for some arguments we will consider cash flows and for others asset values. As long as you keep these relationships in mind, there should be no confusion. The key is to “follow the money!”

Examples Example 7.1 considers a commodity derivative—a bond whose payoff is indexed to gold prices.

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EXAMPLE 7.1: A Bond Indexed to Gold Prices ■

Goldmines Inc. (fictitious name) is a gold mining company. It sells a highly standardized product, pure gold, in the world market. It needs money for exploring, for setting up a mine, for running a mine, and for refining operations. The company’s fortune ebbs and flows with the price of gold.

■

The company decides to raise cash by selling bonds. This is attractive because interest payments are tax deductible. But debt makes a company vulnerable to financial distress—a debt-free company, by definition, can never go bankrupt!

■

An investment banking firm designs a bond with the following payoff on the maturity date T: - A payment of $1,000 - An additional amount tied to gold’s price per ounce, S(T), which is 0 if S (T) < $950 $10 [S (T) − 950] if $950 ≤ S (T)

■

This payoff is given in Table 7.1 and graphed in Figure 7.1. You can generate the first payment synthetically by investing in zero-coupon bonds. The second payoff looks similar to a long call option position: buying ten calls with a strike price of $950 does the trick. Consequently, the cash flows from the traded bond may be obtained synthetically by forming a portfolio consisting of (1) a zero-coupon bond worth 1,000B, where B is today’s price of a zero-coupon bond that pays $1 at maturity, and (2) buying ten European calls, each with a price of c and a strike price of $950 (see Table 7.2).

■

The no-arbitrage principle assures that the replicating portfolio must have the same value as the traded bond. Therefore the bond’s value is (1,000B + 10c). In later chapters, you will learn how to price this call option with the binomial or the Black–Scholes–Merton model.

■

Why sell a security like this? First, it will raise more cash up front. Second, it hedges some of the output price risk that Goldmines Inc. faces. If gold prices are high at maturity, it shares some of its profits—that’s not too bad. And if gold prices decline, only the principal is repaid.

FINANCIAL ENGINEERING

TABLE 7.1: Payoffs for a Traded Bond Indexed to Gold Bond Payoff

Time T (Maturity Date) Cash Flow S(T) < 950

950 ≤ S(T)

1,000

1,000

0 Net cash flow

10[S(T) – 950]

1,000

1,000 + 10[S(T) – 950]

FIGURE 7.1: Payoff Diagram for a Bond Indexed to Gold Payoff Slope = 10

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1,000

0 950

S(T), spot price of gold at maturity

TABLE 7.2: Cash Flows for the Synthetic Bond Indexed to Gold Portfolio

Time T Cash Flow S(T) < 950

950 ≤ S(T)

Long zeros (face value 1,000)

1,000

1,000

Long 10 calls (strike price 950)

0

Net cash flow

1,000

10[S(T) – 950] 1,000 + 10[S(T) – 950]

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Example 7.1 is a commodity-indexed note, which has its return tied to the performance of a commodity (like platinum, gold, silver, or oil) or a commodity index (like Bridge/CRB Futures Index, a commodity futures index that goes back to 1957). Sometimes, the holder can get unlimited benefits from price rises (as in this example), but at other times, the payoffs are capped. Close cousins of these derivatives are equity-linked notes (ELN). An ELN is a combination of a zero-coupon (or a small coupon) bond whose return is based on the performance of a single stock, a basket of stocks, or a stock index (see Gastineau and Kritzman [1996] for definitions and discussions). These come in numerous variations and have fancy trade names like ASPRINs, EPICs, GRIP, SIR, and SUPER. This extra kicker from the equity component makes these securities attractive to buyers, helps the issuers raise more cash up front, and also hedges some of the inherent risks in their business. Example 7.2 shows a hybrid security created to achieve a tax-efficient disposal of a stock that has appreciated in value.

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EXAMPLE 7.2: Hybrid Securities ■

Suppose you are the chief executive officer of Venturecap Co. (a fictitious name) that has bought 1 million shares in a start-up company Starttofly Inc. at $3 per share. The start-up has done very well and is about to go public. Because Venturecap bought these shares through a private placement, securities laws prevent selling those shares for two years. However, you want to dispose of these shares because finance academics report that high prices achieved after an initial public offering usually disappear after six months.

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What should you do? One alternative is for Venturecap to issue hybrid debt (also called a structured note), which is a combination of a bond and a stock. Your investment banker can design premium equity participating securities (PEPS) that allow you to get the benefits of a sale without actually selling Starttofly’s stock.

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How does this work? Suppose the investment banker designs a three-year bond that pays interest at an annual coupon rate of 4 percent per year on the PEPS selling price of $20. The buyer of each PEPS also gets a share of Starttofly after three years. Many investors like the idea of buying Starttofly for $20 in the future and getting the coupon payments over the intermediate years, and the no-arbitrage principle ensures that this security is equivalent to buying a bond and buying the stock.

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Venturecap sells 1 million PEPS and raises $20 million up front. Moreover, it pays capital gains taxes only when it sells the Starttofly stock, which is three years from now, and it gets to deduct the interest payments of $800,000 from its taxes every year.

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Hybrids have been sold in the US, Asia, Europe, and Australia.

APPLICATIONS AND USES OF SWAPS

7.4

An Introduction to Swaps

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Swaps are one of the most successful uses of financial engineering. A swap is an agreement between two counterparties to exchange (“swap”) a series of (usually) semiannual cash flows over the life (tenor or term) of the contract. Salomon Brothers arranged the first swap between International Business Machines Corp. and the World Bank in 1981. A swap contract specifies the underlying currencies, applicable interest rates, payment timetable, and default contingencies. Being over-the-counter (OTC) instruments, swaps do not enjoy the same level of protection as exchange-traded derivatives. You must carefully document them and, if necessary, add collateral provisions to mitigate credit risk (see Chapter 22 for a detailed discussion of swap documentation). As swaps are customized, they come in many flavors and varieties. The simplest swaps go by the fancy name of plain vanilla, whereas more complex varieties are called exotics. Parties in a plain vanilla interest rate swap exchange fixed for floating interest payments, while those in a plain vanilla currency swap exchange fixed payments in one currency for fixed payments in another currency. The libor rate index (see Chapter 2) has traditionally been used as a reference rate for the cash payments on most swap contracts. Nowadays, a large number of swaps are denominated in the European Union’s euro as well as other major currencies like the Japanese yen and the pound sterling. Swaps have flourished because they provide an efficient way of transforming one cash flow to another with very low transaction costs. Our discussion of how the pursuit of lower transaction costs is shaping the global markets is relevant here (see Extension 1.1).

7.5

Applications and Uses of Swaps

Swaps have many financial engineering uses: ■

Transforming loans. A company can transform a fixed interest rate loan into a floating interest rate loan and vice versa through a plain vanilla interest rate swap.

■

Hedging currency risk. A Japanese manufacturer repatriating profits from its US subsidiary can convert dollars into yen through a cross-currency swap and thereby protect itself from currency risk.

■

Altering asset/liability mix. The 1970s showed that a sharp rise in interest rates can bankrupt a savings and loan bank (S&L) specializing in home mortgage lending. A rate hike increases the value of the liabilities (because the deposits must be paid higher interest) but has little immediate impact on the assets (which primarily consist of fixed rate mortgage loans). An S&L can hedge this risk via an interest swap, in which the S&L pays a fixed rate but receives a floating rate. This swap lowers the mismatch in cash flows from both the S&L’s floating rate liabilities (deposits) and fixed rate assets (mortgage loans).

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Creating payoffs that are hard to attain. As there are no exchange-traded derivatives on jet fuel, an airline can hedge its fuel costs by entering into a commodity swap

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that pays the average aviation fuel price computed over a month in exchange for a fixed payment. ■

Avoiding market restrictions. An international investor who is prevented from investing in a country’s stocks by local governmental rules can go around this restriction by entering an equity swap that pays her an amount tied to the return on the local stock index.

7.6

Types of Swaps

As in other financial markets, the swap market has its own financial intermediaries. These swap facilitators (also called swap banks or just banks) either act as a broker (which is becoming less common these days) or perform the role of a dealer. For simplicity, we omit these dealers from our examples. This hardly alters the story. Just think of these dealers as intermediaries who siphon off a few basis points from the cash flows without affecting the basic nature of the transactions.

Interest Rate Swaps

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A plain vanilla interest rate swap exchanges fixed for floating interest rate payments at future dates. As all payments are denominated in the same currency, a notional principal is used to determine the payments, and cash flows are netted. Example 7.3 shows the setup and cash flows from a plain vanilla interest rate swap (see Chapter 22 for a valuation formula).

EXAMPLE 7.3: A Plain Vanilla Interest Rate Swap (Fixed-for-Floating)

Contract Setup ■

Suppose that Fixed Towers Inc. borrowed $100 million for three years at a floating rate of the one-year libor index rate, but it now wants to switch to a fixed interest rate loan. Floating Cruisers Co. raised the same sum and for the same period at a fixed rate of 6 percent but now wants to switch to a floating rate, the one-year libor rate index.

■

They swap. They work out a deal in which Fixed agrees to pay Floating 6 percent and receive the libor rate index on the $100 million notional. This will go on for three years, which is the swap’s tenor or term. The swap is shown in Figure 7.2.

■

Though swaps often have quarterly payments, let’s assume for simplicity that they exchange cash flows at year’s end. As a result of this agreement, Fixed’s cash flows are (– libor + libor – 6 percent =) – 6 percent times the notional, whereas Floating’s cash flows are (– 6 percent + 6 percent – libor =) – libor times the notional. The swap switched each of their existing loans to the other type.

■

It makes no sense to simultaneously exchange $100 million for $100 million. That’s why $100 million is called the notional principal—it exists as a notion, never changes hands, and is only used for computing payments.

TYPES OF SWAPS

The Cash Flows ■

We compute the net payments assuming that the libor index rate takes the values 5, 6, and 7.50 percent, respectively, over years 1, 2, and 3. (1st year, libor = 5 percent) Fixed owes Floating 6 percent times the notional. Floating owes Fixed 5 percent times the notional. Fixed will pay Floating 1 percent of $100 million or $1,000,000 at the end of the first year. (2nd year, libor = 6 percent) Net payment is 0.

FIGURE 7.2: Example of a Plain Vanilla Interest Rate Swap Transforming a Fixed Rate Loan Into a Floating Rate Loan

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Before Swap

Fixed Towers Raised $100 million Cost libor per year

Floating Cruisers Raised $100 million Cost 6 percent per year

libor

6 percent

After Swap (for three years)

Fixed Towers Net cost 6 percent per year

libor

bbalibor 6 percent

Floating Cruisers Net cost libor per year

6 percent

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(3rd year, libor = 7.50 percent) Fixed owes Floating 6 percent times the notional. Floating owes Fixed 7.50 percent times the notional. Fixed receives from Floating 1.50 percent of $100 million or $1.5 million at the end of the third year. Then the contract ends. ■

In reality, counterparties in swaps and other OTC derivatives contracts sign detailed bilateral agreements. Since its founding in 1985, the International Swap Dealers Association (later renamed the International Swaps and Derivatives Association) has been instrumental in developing convenient documents for these contracts.

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Forex Swaps “Rule, Britannia, rule the waves.”2 After attaining supremacy in the early eighteenth century, the Royal Navy was the world’s most powerful navy for over two hundred years, and the British pound reigned supreme. However, the pound lost some of its luster in the mid-twentieth century when the Bretton Woods system established the dollar as the standard currency. The dollar’s dominance is seen in the foreign exchange (forex) market, where most transactions involve exchange of foreign currencies for dollars, and vice versa. The term exchange rate denotes the price of one currency in terms of another. The currencies are quoted either in direct or American terms as dollars per unit of foreign currency (say, $2 per pound sterling) or in indirect or European terms as foreign currencies per unit of dollar (say, £0.5 per dollar). An exchange rate between two currencies other than US dollars is referred to as a cross-rate. Just as you buy an ounce of gold by paying dollars, think of foreign currencies as tradable goods with dollar prices. If the price goes up and you have to spend more dollars to buy one unit of a foreign currency (say, £1 = $3), then we say that the dollar has depreciated. Fewer dollars for a unit of foreign currency (say, £1 = $1) means that the dollar has appreciated. Forex swaps are widely used in forex markets for managing currency risks. A forex swap (or FX swap) is a single transaction with two legs: (1) the contract starts with a spot exchange of one currency for another and (2) it ends with a reverse exchange of these currencies at a subsequent date. Example 7.4 shows the basic idea behind a forex swap.

2

“Rule, Britannia!” is a patriotic British national song, which originated from the poem with the same name written by Scottish poet James Thomson in the mid-eighteenth century.

TYPES OF SWAPS

EXAMPLE 7.4: A Forex (FX) Swap ■

Suppose that Americana Bank has $200 million in excess funds for which Britannia Bank (both fictitious names) has an immediate need. They enter into a forex swap with a tenor of one month.

■

The spot exchange rate SA is $2 per pound sterling in American terms, and its inverse SE = 1/2 = £0.50 per dollar in European terms. Americana gives $200 million to Britannia and receives (Dollar amount × Spot exchange rate in European terms) = $200 × £0.50 per dollar = £100 million today.

■

The annual simple interest rates are i = 6 percent in the US and iE = 4 percent in the United Kingdom. After one month (using a monthly interest rate computation for simplicity), Britannia repays Americana, Dollar amount × Dollar return = $200 × [1 + (0.06/12)] = $201 million and Americana repays Britannia,

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Foreign currency amount × Value of one pound invested for one month = £100 × [1 + (0.04/12)] = £100.3333 million

Instead of beginning immediately, a forex swap may start at a future date. A majority of forex swaps have lives that are measured in weeks rather than months. These versatile instruments have many uses such as (1) managing cash flows in connection with imports and exports, (2) managing borrowings and lendings in foreign currencies, (3) handling foreign exchange balances, and (4) temporarily exchanging excess amounts of one currency for another.

Currency Swaps A plain vanilla currency swap (or a cross-currency swap) is an arrangement between two counterparties involving (1) an exchange of equivalent amounts in two different currencies on the start date, (2) an exchange of interest payments on these two currency loans on intermediate dates, and (3) repayment of the principal amounts on the ending date. Because it’s an OTC contract, the counterparties can modify the terms and conditions at mutual convenience. Some currency swaps skip the exchange of principal and just use them as notional amounts for computing interest. Interest payments are made in two different currencies and are rarely netted.

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Corporations use currency swaps to exchange one currency into another. They do this to repatriate profits and to set up overseas factories. Example 7.5 shows how such a swap works.

EXAMPLE 7.5: A Plain Vanilla Currency Swap (Fixed-for-Fixed) ■

Suppose that Americana Auto Company wants to build an auto plant in the UK and Britannia Bus Corporation (both fictitious names) wants to do the same in the US. To open a factory, you need local currency. Often a manufacturer can more easily raise cash at home because it has relationships with local banks. Americana and Britannia both plan to do this, and seeing a large spread between foreign exchange buying and selling rates, they decide to do the currency conversion via a currency swap. This is a generic fixed-for-fixed currency swap that involves regular exchange of fixed payments over the swap’s life.

The Swap Contract ■

The automakers enter into a swap with a three-year term on a principal of $200 million. The spot exchange rate SA is $2 per pound. Americana raises 100 × 2 = $200 million and gives it to Britannia, which, in turn, raises £100 million and gives it to Americana. Assuming annual payments, Americana pays Britannia 4 percent on £100 million and Britannia pays Americana 6 percent on $200 million at the end of each year for three years. The companies are basically exchanging their borrowings.

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■

The swap ends after three yearly payments, and the principals are handed back. Figure 7.3 shows these cash payments.

Extension 7.1 provides a valuation formula for fixed-for-fixed currency swaps. Currency swaps have several interesting features. First, a swap’s value depends on two sets of interest rates: domestic as well as foreign interest rates of different maturities.3 Second, although our example assumes annual payments, most swaps involve semiannual payments. You can easily add this feature. Notice that Result 7.1 of Extension 7.1 is quite general and does not depend on whether we have annual or semiannual cash flows. Third, the swap value is arbitrage-free. If an errant dealer quotes a different price for the same swap, then you can buy “the cheaper and sell the more expensive swap” and create arbitrage profits. Fourth, anything goes in the swaps markets—these OTC contracts can be designed to satisfy the whims and fancies of the counterparties.

3

Using the covered interest rate parity formula (see Result 12.3), you can eliminate the foreign interest rate and rewrite the formula in terms of the spot and the currency forward rate.

TYPES OF SWAPS

FIGURE 7.3: Plain Vanilla Currency Swap Today: Swap Begins with Exchange of Principal

$200 million Americana Raises $200 million in US at 6 percent

£100 million

Britannia Raises £100 million in UK at 4 percent

$200 million US investors

£100 million UK investors

Intermediate Dates: Interest Payment at the End of First and Second Years

$12 million

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Americana

£4 million

Britannia

$12 million US investors

£4 million UK investors

Maturity Date: Interest Payment and Principal Repayment

$212 million Americana

$212 million US lenders

£104 million

Britannia

£104 million UK lenders

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EXTENSION 7.1: Valuing Fixed-for-Fixed Currency Swaps We expand upon Example 7.5 to develop a valuation formula for currency swaps.

EXT. 7.1 EX. 1: A Valuation Formula for a Currency Swap

Contract Setup ■

■

The situation and the terms guiding this swap are as follows: - Consider the US dollar as the domestic currency and pounds sterling as the foreign currency. - The swap’s tenor T is three years, and it involves annual interest payments. - The spot exchange rate SA is $2 per pound in American terms, which is SE = 1/SA = £0.50 per dollar in European terms. - Americana raises the principal amount L = $200 million at the coupon rate i of 0.06 or 6 percent per year. - Britannia raises an equivalent sum at the coupon rate iE of 0.04 or 4 percent per year. The principal amount LE is calculated by multiplying the dollar principal by the spot exchange rate: L × SE = LE = 200 × 0.50 = £100 million. Americana pays US investors:

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C = Principal (in domestic currency) × Interest rate (on dollar principal) =L×i = 200 × 0.06 = $12 million at the end of each year for three years. - Britannia pays UK investors an annual amount: CE = Principal (in “European” currency) × Interest rate (on foreign currency principal) = LE × iE = 100 × 0.04 = $4 million

The Cash Flows ■

The swap generates the following cash flows (see Figure 7.3): Time 0: Today The swap begins with an exchange of the principals. Americana pays L = $200 million (shown with a solid line arrow) and gets LE = £100 million from Britannia (shown with a broken line arrow).

TYPES OF SWAPS

143

Time 1 (after one year) and Time 2 (at the end of the second year) Interests are paid at year’s end. Americana pays CE of £4 million to Britannia (see solid line arrow), which passes it on to the original British investors (see broken line arrow). Britannia pays C of $12 million to Americana (broken line arrow), which passes it on to the original American investors (solid line arrow). The companies have exchanged their borrowings. Americana has transformed a dollar liability into a sterling liability, while Britannia has done the opposite and accepted a dollar loan. Time 3 (after three years) The swap ends when the companies return their principals and final interest payments. Americana pays £4 million in£4 million in interest and repays the principal of £100 million to Britannia and receives $212 million.

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A Pricing Model ■

Today’s cash flows cancel and do not affect the swap valuation. As Americana will be receiving dollars and paying sterling, we can value this swap to Americana by adding up the present value of all the dollar payments that the company receives and subtracting from this sum the present value (in dollar terms) of all the sterling payments that the company makes.

■

To compute present values, first find the zero-coupon bond (or “zero”) prices in the two countries. They come from the government securities in the US and the UK. The UK government–issued bonds are called gilt securities. Like their American cousins, gilts have a whole range of maturities, semiannual coupon payments, STRIP features, callable issues, and the presence of inflation-protected securities. The UK has even issued consols, which are perpetual bonds that never mature.

■

Denote US zero prices by B(t) and UK zero prices by B(t)E , where t = 1, 2, and 3 years are the times to maturity (see Ext 7.1 Tab. 1).

■

Discount the cash flows by multiplying them by the respective zero-coupon bond prices (Ext. 7.1 Tab. 1 shows how this works). Multiplying the dollar receipts by the US zero prices, we get the following: Present value of Americana’s dollar receipts = PV of dollar interest payments + PV of dollar principal = [(Price of a US zero − coupon bond maturing in one year × Interest payment) + (Price of a zero − coupon bond maturing at time 2 × Cash flow) + (Price of a zero − coupon bond maturing in year 3 × Interest)] + (Price of a zero − coupon bond maturing in three years × Principal) = [B (1) C + B (2) C + B (3) C] + [B (3) L] = (0.95 + 0.89 + 0.82) × 2 + (0.82 × 200) = $195.92 million = PV

(7.1)

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EXT. 7.1 TABLE 7.1: Zero-Coupon Bond Prices in the United States (Domestic Country) and the United Kingdom (Foreign Country) Time to Maturity (in years)

US (Domestic) Zero-Coupon Bond Prices (in dollars)

UK (Foreign) Zero-Coupon Bond Prices (in pounds sterling)

1

B(1) = $0.95

B(1)E = £0.96

2

B(2) = 0.89

B(2)E = 0.91

3

B(3) = 0.82

B(3)E = 0.85

Multiplying the sterling payments by UK zero prices, we get the present value of Britannia’s sterling receipts, which equals Americana’s payments. Consequently, Present value of Americana’s sterling payments = PV of interest payments in pounds sterling + PV of sterling principal = [B (1)E CE + B (2)E CE + B (3)E CE ] + [B (3)E LE ] = (0.96 + 0.91 + 0.85) × 4.00 + 0.85 × 100 = £95.88 million = PVE

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■

Convert this into US dollars by multiplying it by the spot exchange rate: Present value of Americana’s sterling payments in terms of today’s dollars = PVE × SA = 95.88 × 2 = $191.76 million

■

(7.2)

(7.3)

The dollar value of the foreign-currency swap to Americana Auto Company is Value of the currency swap to the domestic investor = (Present value of Americana’s dollar receipts) − [Present value of Americana’s sterling payments (in dollar terms)] = PV − PVE × SA = 195.92 − 191.76 = $4.16 million This means Americana must pay Britannia $4.16 million to enter into the swap.

(7.4)

TYPES OF SWAPS

■

Owing to credit risk, it is prudent to avoid any upfront payment and make this a par swap, which has a zero value. You can do this by tweaking the interest rates until this happens. For example, if you keep i fixed, you will find that iE = 0.047647 makes this a par swap. We restate this as a result (see Ext 7.1 Result 1).

EXT. 7.1 RESULT 1 Valuing a Plain Vanilla Currency Swap A currency swap begins at time 0, when 1. a domestic counterparty (“American”) pays the principal L dollars to a foreign counterparty (“European” or “Foreigner”) and receives an equivalent sum LE in foreign currency 2. American agrees to pay a fixed interest on LE to Foreigner in exchange for a fixed dollar interest on L at the end of each period (times t = 1, 2, . . . , T) 3. the counterparties repay the principals on the maturity date T Then, The value of the swap to American, VSWAP = Present value of dollar receipts − Present value of foreign currency payments (converted dollars into US at the spot rate) = PV − PVE × SA

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145

(7.5)

where PV is the present value of future dollar cash flows received by American, PVE is the present value of future foreign currency payments, and SA is today’s spot exchange rate in American (dollar) terms.The present value of the cash flows is computed as T

PV =

B (t) C + B (T) L

(7.6)

B (t) i + B (T) L ]

(7.7)

∑ t=1

T

=

[∑ t=1

where B(t) is today’s price of a US zero-coupon bond that pays one dollar at time t and C is the coupon or the interest on the dollar principal that is paid at times t = 1, 2, . . . , T (computed as dollar principal times the simple interest rate or L × i). PVE is similarly computed by attaching a subscript E and by replacing domestic by foreign currency cash flows and zero-coupon bond prices in the preceding formula.

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Commodity and Equity Swaps Example 7.6 illustrates a simple commodity swap.

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EXAMPLE 7.6: Hedging Airline Fuel Costs with a Commodity Swap ■

Aviation fuel is a major cost for airlines. Some airlines hedge this price risk by entering into a commodity swap in which they pay a fixed price and receive the average aviation fuel price computed over the previous month.

■

Suppose HyFly Airlines (fictitious name) needs 10 million gallons of aviation fuel per month, which it buys on a regular basis from the spot market. Seeing a huge rise in oil (and aviation fuel prices) in recent years, HyFly is scared that a continued run-up will ruin the airline, and it decides to hedge its exposure by entering into a commodity swap.

■

The swap is structured as follows (see Figure 7.4 for a diagram of this swap): - The notional is 10 million gallons of aviation fuel. - The swap has a two-year term, and the payments are made at the end of each month. - HyFly pays the dealer a fixed price of $1.50 per gallon or 10,000,000 × 1.50 = $15 million each month. - The dealer pays a floating price that is the average spot price of fuel during the previous month. - The payments are net.

■

Suppose last month’s average price is $1.60 per gallon. - Then the dealer pays 10,000,000 × 1.60 = $16 million at the end of the this month. - As the payments are netted, the airline receives $1 million. - This extra payment helps alleviate the airline’s higher aviation fuel price paid in the spot market.

■

Suppose oil prices fall unexpectedly and last month’s average fuel price was $1.30. The savings that HyFly generates from the lower price have to be surrendered to the dealer, who must be paid $2 million$2 million by the terms of the contract.

This commodity contract is hard to price because it involves an average price (see Jarrow and Turnbull [2000] for a valuation formula for this swap). Equity swaps are similar to the swaps previously discussed. The simplest equity swaps involve counterparties exchanging fixed rate interest payments for a floating payment that is tied to the return on a stock or a stock index. The payments are computed on a notional and exchanged at regular intervals over the swap’s term. The index may be a broad-based index (such as the Standard and Poor’s 500) or a narrowly defined index (such as for a specific industry group like biotechnology). Equity swaps allow money managers to temporarily change the nature of their portfolios and bet on the market. Suppose that we find two portfolio managers who hold completely opposite views about the direction of the stock market. They can enter into an equity swap in which the bullish manager receives a floating rate tied to the market index and pays the bearish manager a fixed interest rate.

TYPES OF SWAPS

FIGURE 7.4: Hyfly Airline Hedging Fuel Price Risk For 2 Years

$1.50 per gallon Hyfly Airline

Swap Dealer Average spot price

Aviation fuel

Spot price of aviation fuel

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Spot Market

Credit Default Swaps Credit default swaps (CDS) have been hailed during good times as a “wonder of modern finance” and vilified during the financial crisis of 2007 as a “‘Ponzi scheme’ that no self-respecting firm should touch” (see “Credit Derivatives: The Great Entangling,” Economist, November 6, 2008). Designed during the 1990s (by J. P. Morgan & Co.), CDS combine the concepts of insurance and swaps to create a contract that plays a useful economic role. It enables the holder of a risky bond to sell the credit risk embedded in the bond to a counterparty willing to bear it. Let’s consider how a CDS works. A CDS is a term insurance policy on an existing corporate or government bond. A typical CDS matures in one to five years and has a notional amount equal to the face value of the bond being insured, with most CDS being in the $10 to $20 million range. The seller of a CDS (the insurer) receives a periodic premium payment (usually quarterly) for selling the insurance. If a credit event occurs—the bond defaults—then the buyer of the CDS (the insured) receives the face value of

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the debt from the seller but hands over the bond in return.4 The CDS contract ends if a credit event occurs. The premium payment is quoted as a CDS spread (say, 100 basis points per annum). The actual dollar payment is the prorated spread (adjusted for the length of the payment period) times the notional. The CDS spread is quoted in the market so that no cash is exchanged when the CDS is written. This implies that the CDS has zero value at initiation. Example 7.7 illustrates the working of a CDS.

EXAMPLE 7.7: Hedging Default Risk ■

A pension fund, the Worried Lenders Fund (a fictitious name) has purchased a bond issued by Candyfault Enterprises (CE) (another fictitious name). The bond has five years until it matures, pays a 5.5 percent annual coupon, and has a notional of $5 million. The pension fund manager is concerned that Candyfault might default on its interest payments over the next year owing to the current recession. She wants to hedge its default risk but does not want to sell the bond.

■

To hedge the default risk on CE over the next year, it buys a one-year CDS on CE with a notional of $5 million. The CDS quoted spread is 400 basis points per year, paid quarterly. The Worried Lenders Fund buys this CDS.

■

The quarterly premium payment made by the Worried Lenders Fund is 0.04 × (1/4) × $5 million = $50, 000

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■

The CDS seller gets this payment quarterly until the CDS expires or until CE defaults on its bond, whichever comes first. In the event of default, the CDS seller pays the Worried Lenders Fund $5 million, and it takes possession of the CE bond.

Pricing of this simple contract, however, is a difficult task. CDS are also written on asset backed securities, called ABS CDS, but their cash flows are somewhat different (see Jarrow 2011).

7.7

Summary

1. We emphasize a build and break approach, in which derivatives can be broken down into component parts or combined with other derivatives to create new ones. 2. Once the building blocks are understood, one can venture into financial engineering, a new discipline that applies engineering methods to financial economics. 3. A swap is an agreement between two counterparties to exchange a series of cash payments over its life. Swaps are successful vehicles for transforming one kind of 4

This is the case of physical delivery. If cash delivery occurs instead, the swap buyer receives the difference between the face value of the debt and the market price of the defaulted debt issue (usually determined through an auction).

QUESTIONS AND PROBLEMS

cash flow to another. Swaps are OTC contracts that can be customized to the counterparties. The simplest kinds of swaps have the following structure: a. A plain vanilla interest rate swap exchanges fixed for floating rate payments. The principal, called the notional principal, never changes hands but is used only for computing cash flows. Payments are netted because the cash flows are denominated in the same currency. b. A forex swap involves a spot exchange of foreign currencies that is followed by a reverse exchange of equivalent amounts (principal plus relevant interest) at a future date. c. A plain vanilla currency swap (or a cross-currency swap) is an arrangement between two counterparties involving (1) an exchange of equivalent amounts of two different currencies on the starting date, (2) an exchange of interest payments on these two currency loans on intermediate dates, and (3) repayment of the principal on the ending date. d. A commodity swap involves an exchange of an average price of some notional amount of a commodity in exchange for a fixed payment. e. An equity swap exchanges fixed rate interest payments for a floating payment that is tied to the return on a stock or a stock index. f. A credit default swap is a term insurance policy on a bond.

7.8

Cases

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Times Mirror Company PEPS Proposal Review (Harvard Business School Case

296089-PDF-ENG). The case examines the design of a premium equity participating security for tax-efficient disposal of an appreciated common stock. Privatization of Rhone-Poulenc—1993 (Harvard Business School Case 295049-

PDF-ENG). The case illustrates how a recently privatized French company can use financial engineering to design an investment plan for employees that would enhance their ownership of company shares. Advising on Currency Risk at ICICI Bank (Harvard Business School Case 205074-

PDF-ENG). The case studies how a large Indian bank can hedge risks from multiple interest rates and currencies by using a variety of derivatives like swaps, options, and futures contracts on interest rates and currencies.

7.9

Questions and Problems

7.1. Briefly describe a plain vanilla interest rate swap. Is the notional principal paid

out at the swap’s beginning and maturity? 7.2. Briefly describe a currency swap. Is the notional principal paid out at the swap’s

beginning and maturity? 7.3. How does a plain vanilla interest rate swap differ from a currency swap?

The next two questions use the following information: American Auto enters into a plain vanilla currency swap deal with European Auto. American raises $30 million at an 8 percent fixed interest rate, while European raises an equivalent amount of €33 million at 9 percent. They swap

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these two loans. The swap lasts for three years, and the payments take place at the end of each year. 7.4. What is the principal? Does it change hands at the beginning and end of the

swap? 7.5. Calculate the gross payments involved and indicate who pays what in this swap

deal. The next two questions use the following information: Esandel Bank enters into a plain vanilla interest rate swap with a swap facilitator Londoner Inc. Esandel pays a fixed amount of 8 percent per year to Londoner, which in turn pays a floating amount libor + 1.50 percent. The principal is $15 million. The swap lasts for five years, and the payments take place at the end of each year. 7.6. What is the notional principal? Does it change hands at the beginning and end

of the swap? 7.7. Who is in the “receive fixed” situation? Who is in the “pay-fixed” situation? 7.8. Calculate the net payments involved and indicate who pays what in this swap

deal if the libor takes on the values 7.00 percent, 6.50 percent, 7.00 percent, 7.50 percent, and 6.00 percent at the end of the first, second, third, fourth, and fifth year, respectively. Consider the swap in Extension 7.1:

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- The automakers enter into a swap with a three-year term on a principal of $200 million. - The spot exchange rate SA is $2 per pound. Americana raises 100 × 2 = $200 million and gives it to Britannia, which in turn raises £100 million and gives it to Americana. - Americana pays Britannia 4 percent per year on £100 million and Britannia pays Americana 6 percent per year on $200 million at the end of each year for three years. Now assume that the companies make payments every six months: the swap ends after six semiannual payments, and the principals are handed back after three years. 7.9. Using zero-coupon bond prices (maturing every six months) given in

Table 7.3, compute the value of this swap. 7.10. How can one use a currency swap to hedge currency risk? 7.11. How can one use an interest rate swap to change a fixed rate loan into a floating

rate loan? 7.12. How can a savings and loan bank use an interest rate swap to match its long-

term fixed rate investments with the risks of its short-term floating rate demand deposit obligations?

QUESTIONS AND PROBLEMS

TABLE 7.3: Zero-Coupon Bond Prices in the United States (Domestic Country) and the United Kingdom (Foreign Country) Time to Maturity (in years)

US (Domestic) Zero-Coupon Bond Prices (in dollars)

UK (Foreign) Zero-Coupon Bond Prices (in pounds sterling)

0.5

B(0.5) = $0.99

B(0.5)E = £ 0.98

1

B(1) = 0.97

B(1)E = 0.96

B(1.5) = 0.95

B(1.5)E = 0.93

B(2) = 0.93

B(2)E = 0.91

B(2.5) = 0.91

B(1.5)E = 0.88

B(3) = 0.88

B(3)E = 0.85

1.5 2 2.5 3

7.13. What is an equity swap? What is a commodity swap? 7.14. Suppose an investor is precluded from investing in a country’s stocks by

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government regulations but can invest in the country’s government bonds. Design a swap to overcome these regulations. 7.15. What is a forex swap? Explain how it works. 7.16. In Chapter 2, we described how the US Treasury STRIPS worked. Explain

how Wall Street firms had financially engineered a product similar to STRIPS. 7.17. What is a callable bond? When would it be used? 7.18. What is a putable bond? When would it be used? 7.19. What is a convertible bond? How would you break it down into simpler parts

and price it? 7.20. Check the Internet or other sources and answer the following questions: a. Briefly describe the Troubled Asset Relief Program (TARP) program of the

US government. b. What was the derivative used in the program? Why would the US

government trade such a “toxic thing” as a derivatives contract? c. How would you approach the pricing of the security used in the TARP? d. Discuss the success of the program to date.

151

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II

Forwards and Futures

CHAPTER 8

Forwards and Futures Markets

CHAPTER 9

Futures Trading

CHAPTER 10

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Futures Regulations

CHAPTER 12

The Extended Cost-of-Carry Model

CHAPTER 13

Futures Hedging

CHAPTER 11

The Cost-of-Carry Model

8 Forwards and Futures Markets Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

8.1 Introduction 8.2 Applications and Uses of Forwards and Futures 8.3 A Brief History of Forwards and Futures Early Trading of Forward and Futures-Type Contracts

8.4 Futures Contract Features and Price Quotes Commodity and Financial Futures Contracts The Gold Futures Contract Gold Futures Price Quotes The Exchange and Clearinghouse

US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926

8.5 Commodity Price Indexes

Recent Developments, 1970 Onward

8.6 Summary

EXTENSION 8.1 Doomsters and Boomsters

8.7 Cases 8.8 Questions and Problems

APPLICATIONS AND USES OF FORWARDS AND FUTURES

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8.1

Introduction

In 1688, the world’s first futures exchange, the Dojima Rice Exchange, opened its doors. It was located in Osaka, which was Japan’s commercial center at the time known as the “country’s kitchen.” In those days, if you controlled rice, then you controlled the Japanese economy. The Dojima Exchange devised a fairly advanced system of trading that has many commonalities with the way futures trade today. A day’s trading time was determined by a firebox system. The market would open in the morning with the lighting of a wick in a hanging wooden box, and it would close when the wick was completely burned down. If the wick was extinguished early, then all trades of that day were canceled. You can imagine how losing traders tried to puff out the fire early, while winning traders and exchange officials attempted to foil such plans. Raw emotions would sometimes get expressed through brawls and fistfights (see Alletzhauser 1990; West 2000). As this story relates, emotions (and stakes) can run high at futures exchanges, the workings of which are described in this chapter. However, as floor trading is being phased out, action these days takes place behind electronic terminals in airconditioned rooms. We take up forward and futures contracts from where we left off in Chapter 4. We discuss the usefulness of forward and futures contracts and present a brief history. Although futures-type contracts have traded in many times and places, the evolution of the modern futures contract really began in mid-nineteenth-century Chicago and was completed in about seventy-five years. After decades of relative calm, the 1970s saw a period of astonishing changes in futures markets. These developments can be classified into two categories: (1) the introduction of new futures contracts, and (2) the opening of new exchange and the automation of trading. Finally, we examine a gold futures contract and study how futures prices get reported in the financial press and quoted by vendors.

8.2

Applications and Uses of Forwards and Futures

As forward and futures markets are growing by leaps and bounds (and huge securities markets don’t exist without good reason), they must be serving a useful role in the economy. And this growth is not just in futures on commodities like corn and gold. The 1970s saw the introduction of financial futures (“financials”), which are futures contracts based on financial assets (such as foreign currencies) and on economic or financial variables (such as interest rates). These contracts have become very popular. What is the utility of forward and futures contracts including those written on financial assets and financial variables? Futures and forwards take investors far beyond the realm of ordinary stocks and bonds and, in the language of Star Trek, “where no man has gone before.” Consider the following observations: 1. By purchasing forwards and futures in advance, a trader can smooth unexpected future price fluctuations that can come from demand–supply mismatches.

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2. Forwards are useful for acquiring a commodity at a fixed price at a later date. Futures are seldom used for trading assets because of the costs involved with physical delivery. However, their features make them excellent tools for managing price risk. 3. Forwards and futures help make the market more “complete.” In a complete market, sufficient securities trade such that investors can construct portfolios to obtain all possible probability distributions over future payoffs. 4. They help traders to speculate. Speculators have a beneficial presence in most futures markets. Speculators increase liquidity and enhance market efficiency, but destabilizing speculation can also take place, and we look at several manipulation cases in Chapter 10. 5. They allow investors to leverage their capital and hold large trading positions without tying up cash. However, like speculation, leverage cuts both ways and can sometimes do more harm than good. For example, futures bets destroyed Barings PLC, a venerable British bank that had existed for over two hundred years. 6. The process of trading forwards and futures generates useful information about future price expectations called price discovery, an important function discussed in greater detail in chapter 10.

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8.3

A Brief History of Forwards and Futures

Forward trading began early in the dawn of civilization (see Table 8.1) and thrives to this day. The history of forwards and exchange-traded futures may be divided into three phases: 1. Early trading of forward and futures-type contracts. Forward contracts evolved separately in many parts of the world. Sometimes they traded in centrally located markets with standardized trading rules and contract terms that prompted some scholars to label them as futures contracts. 2. Evolution of the modern futures contract. Today’s futures contracts started taking shape in the United States during the nineteenth century and arrived at its present form over the next seventy-five years. 3. Developments since the 1970s. An astonishing variety of futures were conceived and marketed during the last three decades of the twentieth century.

Early Trading of Forward and Futures-Type Contracts Forward contracts have been used to reduce risk for eons. Forwards traded in India around 2000 BC, in ancient Greece and Rome, in medieval Europe, and in Japan during the seventeenth century (see Table 8.1). Records indicate that Amsterdam grain dealers used both futures and options in the 1550s and the decades that followed to hedge trade goods shipped far distances.

A BRIEF HISTORY OF FORWARDS AND FUTURES

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TABLE 8.1: Early History of Trading Forward and Futures-Type Contracts Year

Development

2000 BC and onward

Forward contracts traded in Greece, India, and the Roman Empire

Eleventh and twelfth centuries AD and onward

Medieval fairs

1531

The first commodities exchange in Antwerp, Belgium

1550s onward

Futures and options contracts traded in Amsterdam, Netherlands

1571

Royal Exchange for trading commodities in London, England

1688

Dojima Rice Exchange in Osaka, Japan

One of the earliest commodities exchanges was the Royal Exchange of London, which was inaugurated by Queen Elizabeth I in 1571. Today the original location houses an upscale shopping mall. The renowned London Metal Exchange and the London Stock Exchange both trace their ancestry back to the Royal Exchange. The world’s oldest stock exchange, as noted in Chapter 2, opened in Amsterdam in 1602. Soon futures and options contracts started trading. This was a time when the tulip was gaining popularity in Holland. A few decades later, the Dutch got involved in an extraordinary speculative fervor over rare tulip bulbs, and futures and options were devised to help the speculating public. As with other price bubbles, the famous tulip bulb price bubble ended when prices came crashing down and derivatives trading dried up. As noted at the beginning of this chapter, Osaka’s Dojima Rice Exchange opened its doors in 1688. It is considered the world’s first organized futures market because the contracts were standardized, had fixed life, required margin payments, and involved clearinghouses. The exchange operated for centuries and was closed down in 1939 because of wartime controls.

US Futures Exchanges and the Evolution of the Modern Futures Contract, 1848–1926 Despite interesting developments elsewhere, modern futures trading really began in Chicago in the mid-nineteenth century with the establishment of the Chicago Board of Trade (CBOT or CBT). Why Chicago? Located on the windswept shores of Lake Michigan, the “Windy City” is close to the fertile farmlands of the great Midwest. Nineteenth-century Chicago became America’s major transportation and

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TABLE 8.2: Evolution of Today’s Futures Contract (1848–1926) Year

Development

1848

The CBOT was established to standardize grains trading

1849–50

To arrive contracts started trading at the CBOT

1865

General rules were developed at the CBOT that standardized futures contracts; traders were required to post margins

1874

The Chicago Produce Exchange was formed to trade farm products; it became the Chicago Butter and Egg Board in 1898, and in 1919, it became the CME, with its own clearinghouse

1877

Speculators were allowed to trade at the CBOT

1882

NYMEX was established

1925

A clearinghouse was established at the CBOT

distribution center for agricultural produce. Farmers shipped their grains to Chicago, much of which then moved eastward to where most Americans lived. But agriculture is a seasonal business. Prices crashed at harvest times as farmers flooded Chicago with grains and rose again as the grain was used and became scarce. A need was felt for a central marketplace to smooth such demand–supply imbalances. The futures market was devised to fill this void. To standardize the quality and quantity of grains traded, eighty-two merchants founded the CBOT in 1848. Spot market trading began immediately, and more developments followed. Table 8.2 shows how US futures trading evolved to reach its present-day perfection. In the mid-1860s, the CBOT streamlined trading and further standardized contracts in terms of size, quality of commodities, delivery dates, and places. The contracts came to be called futures. Most important, the CBOT also required performance bonds called margins from both buyers and sellers. This eliminated counterparty risk. In 1877, the CBOT allowed speculators to trade. Prior to this time, only producers or purchasers could trade. In contrast, speculators trade for profits and not for hedging purposes. Meanwhile there was a need to organize the trading of butter, eggs, poultry, and other farm products. Merchants trading these commodities founded the Chicago Produce Exchange in 1874; this became the Chicago Butter and Egg Board in 1898 and emerged as the Chicago Mercantile Exchange (CME or the Merc) in 1919. It became widely known for trading traditional futures contracts on dairy, meat, and poultry products. A group of merchants founded the Butter and Cheese Exchange of New York, renamed the New York Mercantile Exchange (NYMEX) in 1882. Another innovation came when clearinghouses were established by the CME after its reorganization in 1919, and by the CBOT in 1925. As noted earlier, in exchangetraded derivatives markets, a clearinghouse plays the crucial role of clearing a trade by matching the buyer and seller, recognizing and recording trades, and (nearly)

A BRIEF HISTORY OF FORWARDS AND FUTURES

159

eliminating counterparty default risk. These developments almost created the futures contract as we know it today.

Recent Developments, 1970 Onward Futures markets have undergone significant changes since 1970 (see Table 8.3). We classify these changes into two categories: (1) the introduction of new futures contracts, and (2) the opening of new exchanges and the automation of trading.

NEW FUTURES CONTRACTS In 1972, the CME created the International Monetary Market (IMM) division for trading forex futures. The seven foreign currency futures that started trading at the IMM became the first successful financial futures traded. But exchanges don’t always succeed. Credit for the first currency futures exchange goes to the International Commercial Exchange, an exchange founded by the members of the New York Produce Exchange in 1970 but that closed its doors in 1973. Its failure has been attributed to several causes: (1) bad timing (as it was founded when the Bretton Woods agreement was still in force), (2) limitations of a small exchange, (3) small contract size, (4) a failure to adequately promote itself, (5) high margin requirements, and (6) high brokerage fees (see “New Game in Town,” Wall Street Journal, May 16, 1972).

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TABLE 8.3: Milestones in the History of Futures Trading since 1970 Year

Development

1972

The first successful financial futures, foreign currency futures, started trading at the newly created IMM at the CME

1973

The CBOE was founded by members of the CBOT

1974

US Congress passed the Commodity Futures Trading Commission Act that created the CFTC

1975

The first interest rate futures, GNMA futures contracts, started trading at the CBOT

1977

Treasury bond futures began trading at the CBOT

1981

The first cash-settled contract, the eurodollar futures, started trading at the CME

1982

The NFA, a self-regulatory organization made up of firms and people who work in the futures industry, was established

1982

Options on US Treasury bond futures started trading at the CBOT

1982

The first index futures contract, the Value Line index futures, started trading at the Kansas City Board of Trade; within months, the S&P 500 stock index futures started trading at the CME

1984

The first international futures link, the CME/SIMEX mutual offset trading link, was established

1992

A postmarket global electronic transaction system, Globex, was launched by the CME and Reuters

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CHAPTER 8: FORWARDS AND FUTURES MARKETS

Novel products and new markets generate a need for greater regulatory oversight. In 1974, the US Congress passed the Commodity Futures Trading Commission (CFTC) Act, which created the CFTC as the “federal regulatory agency for futures trading.” In 1982, the National Futures Association (NFA), an industry body, was established. NFA does a slew of self-regulatory activities, which (remember from Chapter 1) enhances the reputation of the futures markets and reduces the need for greater federal oversight. In the 1970s, oil shocks and other supply-side disturbances hiked up the inflation rate, causing interest rates to be more volatile. In 1975, the CBOT launched the Government National Mortgage Association (GNMA) futures, which protected mortgage holders from interest rate risk. This was the first interest rate derivative. The Ginnie Mae futures contract was the spark that ignited the creation of the vast global market for interest rate derivatives, one of which was the eurodollar futures introduced by the CME in 1981. Eurodollar futures were the world’s first cash-settled futures contracts. Normally, when a futures contract is carried to delivery, the short collects the futures price and delivers the underlying commodity. A cash-settled futures contract has no such physical delivery provision. If taken to maturity, a cash payment equal to the difference between the settlement price at the contract’s end and the previous day’s settlement price closes out the contract. Without cumbersome delivery, it became possible to design an extraordinary range of derivatives based on notional variables like indexes, interest rates, and other intangibles. The year 1982 saw the introduction of options both on futures and on stock index futures. In Chapter 5, we introduced options on spot, which are options contracts based on an underlying asset or a notional variable. Equity options on blue chip stocks like Ford and International Business Machines are its simplest examples. By contrast, options on futures or futures options have a futures contract as the underlying. This market started in 1982 with US Treasury bond futures. Most futures options now trade side by side with their underlying futures contracts on their respective exchanges. Index futures came as a natural next development. These are cash-settled futures contracts that have an index as the underlying variable. In 1982, the Kansas City Board of Trade it got regulatory approval to trade futures on the Value Line Index. This is an average of over 1,700 stocks tracked by Value Line Incorporated, a well-known provider of investment advice. Months later, the CME introduced the Standard and Poor’s 500 index futures (S&P 500), which soon overtook the former. Index futures are very popular for implementing different trading strategies, including hedging, speculation, and index arbitrage. Subsequent years saw the introduction of an assortment of new contracts. Futures contracts on interest rates in major currencies like dollars (eurodollars), the euro, the Japanese yen, and the pound sterling are offered by major exchanges including the CME Group. Moreover, they trade on short-term interest rate benchmarks, which are often determined by a method similar to the libor index rate (see Chapter 2) and use the suffix IBOR or BOR (for “interbank offered rate”). Examples include interest rate futures on BUBOR in Budapest, HIBOR in Hong Kong, KORIBOR

FUTURES CONTRACT FEATURES AND PRICE QUOTES

in Korea, MOSIBOR in Moscow, and STIBOR in Stockholm. The emergence of the European Community has also created the need for euro-based futures. Just open the Wall Street Journal or go to the website of one of the futures exchanges to get an idea of this diversity.

NEW EXCHANGES AND THE AUTOMATION OF TRADING In terms of trading volume, some of the world’s largest futures and options exchanges have opened in recent years. There are futures exchanges even in countries with emerging financial markets. In all, nearly a hundred futures exchanges operate today. Many of them are modeled after US futures exchanges, and they even relied on American technical expertise to be set up. The European Union’s motto “United in Diversity” is an apt description of today’s financial world: numerous exchanges with diverse product lines in distant lands have proliferated with customers linked through a single electronic terminal. Recent years have seen a trend toward extended trading hours. As discussed in Chapter 3, IT advances have revolutionized securities trading by making innovations like electronic exchanges, online brokerage accounts, and network linkage fairly commonplace.

8.4

Futures Contract Features and Price Quotes

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This section discusses a futures contract’s specifications and how prices and trades are recorded.

Commodity and Financial Futures Contracts Futures contracts are divided into those on commodities or financial futures. Commodity futures usually refer to futures contracts on commodities like cattle, coffee, copper, and corn. Financial futures refer to futures contracts on financial assets like bonds, currencies, or even on notional variables like stock indexes and interest rates (see Table 8.4).1 Greater connectivity among exchanges has considerably expanded the menu of choices available for trading futures. Most futures have similar features, especially those trading on US exchanges. As “seeing one is seeing them all,” we select a popular contract and examine its specifications. Gold is the natural choice since gold futures trade on several US exchanges and on at least ten foreign ones.

The Gold Futures Contract Let us examine some features of the gold futures contract that trades on the COMEX division of the CME Group (see Example 8.1).2 1

Legally the term commodity has a broader definition that includes nearly all goods and services (see Title 7 [Agriculture], Chapter 1 [Commodity Exchanges], Section 1a of the US Code), and under this legal definition, financial futures are considered commodity futures. 2

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EXAMPLE 8.1: The Gold Futures Contract Traded on the COMEX Division of the CME Group ■

A trading or ticker symbol identifies the futures contract. The gold futures contract has the ticker symbol GC.

■

A trading month identifies the contract’s delivery month. At any time, GC contracts trade with “delivery during the current calendar month; the next two calendar months; any February, April, August, and October falling within a 23-month period; and any June and December falling within a 72-month period beginning with the current month.”3 Table 8.5 gives actual price quotes for these contracts and shows some months that trade. The choice of trading months is crucial for many commodities. As gold is produced steadily throughout the year, it makes sense to have delivery months spaced at regular intervals throughout the year. Futures on agricultural commodities often have delivery months clustered around harvest times.

■

The trading unit or contract size (also called even lot) is one hundred troy ounces per contract. Some commodities including gold have futures contracts of multiple sizes.

■

Trading hours In both the CME Globex (electronic trading system) and CME ClearPort (system for clearing over-the-counter trades), trading takes place from 24 hours a day (with a daily break from 4 PM to 5 PM Chicago time), seven days a week. Floor trading for gold futures has been phased out.

■

The price quotation is in US dollars and cents per troy ounce. If the quoted price is $500, then the position size for one contract is 500 × 100 = $50,000.

■

The minimum price fluctuation or tick size measures the minimum price jump. GC has a tick size of 10 cents ($0.10) per troy ounce or $10 per contract. For example, if the gold futures price is $500.00, the next higher price would be $500.10, the next would be $500.20, and so on. The long position gains and the short position loses by $10 for each of these jumps. If the price falls from $500.00 to $498.50, then the long loses $150 per contract.

■

The maximum daily price fluctuation or daily price limit is the maximum price change allowed in a day. Some commodities have price limits (see the relevant exchange websites for details).

■

The last trading day is the third to last business day of the delivery month. This is the last day on which a GC contract maturing in that month can trade.

■

Delivery is the traditional way of ending a futures contract. Gold delivered against a GC contract must bear a serial number and an identifying stamp of a refiner approved and listed by the exchange. Delivery must be made from a depository that has been licensed by the exchange.

■

Besides delivery, a contract may end with an exchange of physicals (EFP) for the futures contract. In this case, the buyer and the seller privately negotiate the exchange of a futures position for a physical position of equal quantity. EFPs take place at locations that are away from the exchange floor.

■

The delivery period is the time period during which delivery can occur. The first delivery day is the first business day of the delivery month, while the last delivery day is the last such day. Like most futures, the GC seller decides if a delivery should take place, and if so, when to start the process and what grade to deliver. This makes sense because historically, sellers were hedgers who held the physical commodity for sale.

FUTURES CONTRACT FEATURES AND PRICE QUOTES

TABLE 8.4: Examples of Futures Contracts That Trade in the United States Category

Examples of Commodities

COMMODITY FUTURES Grains and oilseeds

Corn, soybean, wheat, canola, . . .

Livestock and meat

Cattle, pork belly, hogs, . . .

Food and fiber

Coffee, orange juice, sugar, cotton, . . .

Metals

Copper, gold, platinum, silver, . . .

Energy

Gasoline, heating oil, natural gas, electricity, . . .

Other products

Ammonia, lumber, plywood, . . .

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FINANCIAL FUTURES Interest rates

Canadian bonds, euribor, eurodollars, Treasury securities, . . .

Currencies

Euro, peso, pound, Swiss franc, . . .

Indexes

Dow, Nasdaq, Nikkei, S&P 500, . . .

■

Trading at Settlement allows traders to trade any time during trading hours but at a delivery price that is the settlement price determined at the day’s close. This is only allowed for certain months.

■

Regarding grade and quality specifications, in case of delivery, the seller must deliver 100 troy ounces of gold, subject to various specifications regarding assaying fineness, the form in which it can be delivered, and acceptable refiners (details are available from the exchange on request.)

■

The position limit restricts the number of contracts that a speculator can hold in particular commodities (see the relevant exchange websites for details). Position limits try to prevent traders from manipulating by accumulating huge long positions and then squeezing short sellers. Several forward and futures market manipulation cases discussed in chapter 10 show that this risk is real.

■

Margins are security deposits required to open and maintain futures positions. A nonmember opening a speculative position has to keep an initial margin (security deposit needed to open a futures position) per contract. The maintenance margin is the amount that if the account value falls to this level, the trader has to supply more funds to bring the margin back to the initial margin. For the exact dollar margins required, see the relevant exchange websites.

3

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TABLE 8.5: COMEX Gold Futures Prices at Market Close on Thursday, August 24, 2007

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GOLD (Comex Division, NYMEX) 100 troy oz.; $ per troy oz.

Last

Open High

Open Low

High

Low

Aug 2007

668.10

n/a

662.10

668.80

662.10

Sep 2007

667.00

659.60

659.60

667.00

659.60

Oct 2007

671.20

665.00

664.00

673.10

661.00

Nov 2007

n/a

n/a

n/a

n/a

n/a

Dec 2007

678.00

671.20

670.90

679.00

666.70

Feb 2008

683.00

n/a

673.90

683.10

673.00

April 2008

679.40

679.40

679.40

679.40

679.40

June 2008

694.00

685.00

685.00

694.00

685.00

Aug 2008

n/a

n/a

n/a

n/a

n/a

Oct 2008

n/a

n/a

n/a

n/a

n/a

Dec 2008

710.00

n/a

701.30

710.00

701.30

Feb 2009

n/a

n/a

n/a

n/a

n/a

April 2009

n/a

n/a

n/a

n/a

n/a

June 2009

n/a

n/a

n/a

n/a

n/a

Dec 2009

n/a

n/a

n/a

n/a

n/a

June 2010

n/a

n/a

n/a

n/a

n/a

Dec 2010

n/a

n/a

n/a

n/a

n/a

June 2011

n/a

n/a

n/a

n/a

n/a

Dec 2011

n/a

n/a

n/a

n/a

n/a

June 2012

n/a

n/a

n/a

n/a

n/a

Gold Futures Price Quotes Wire news services, websites, and many major daily newspapers carry futures price quotes. As there are some differences in how they report, we discuss price quotes for the COMEX gold futures. Table 8.5 gives these prices for Thursday, August 24, 2007, from the NYMEX website (this contract currently trades on the CME Group’s COMEX division; quotes are available online4 ). Newspapers provide basic

FUTURES CONTRACT FEATURES AND PRICE QUOTES

165

8.6: COMEX Gold Futures Prices at Market Close on Thursday, August 24, 2007

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mex Division, NYMEX) 100 troy oz.; $ per troy oz.

Most Recent Settle

Change

Open Interest

Estimated Volume

Last Updated

668.00s

+0.10

261

204

8/24/2007 2:57:39 pm

668.70s

−1.70

234

41

8/24/2007 2:58:42 pm

671.50s

−0.30

35815

2069

8/24/2007 3:32:29 pm

n/a

0

n/a

n/a

n/a

677.50s

+0.50

188352

59490

8/24/2007 3:41:59 pm

683.40s

−0.40

16264

659

8/24/2007 1:49:42 pm

689.10s

−9.70

17710

9

8/24/2007 1:41:15 pm

694.70s

−0.70

14606

37

8/24/2007 1:42:53 pm

700.10s

0

7541

47

8/24/2007 1:41:15 pm

705.50s

0

1469

n/a

8/24/2007 1:41:15 pm

710.90s

−0.90

15978

181

8/24/2007 1:41:15 pm

716.60s

0

11076

n/a

8/24/2007 1:41:15 pm

722.20s

0

1630

n/a

8/24/2007 1:41:15 pm

727.80s

0

10983

n/a

8/24/2007 1:41:15 pm

744.90s

0

2566

n/a

8/24/2007 1:41:15 pm

762.30s

0

2569

n/a

8/24/2007 1:41:15 pm

780.10s

0

2585

n/a

8/24/2007 1:41:15 pm

798.10s

0

1724

n/a

8/24/2007 1:41:15 pm

816.50s

0

1035

n/a

8/24/2007 1:41:15 pm

835.90s

0

0

n/a

8/24/2007 1:41:15 pm

information like where the contract trades, the size of the contract, and how prices are quoted. This redundant information is not reported on the NYMEX website. The first column lists the delivery months. The last trade price is reported next: this was $678.00 for the December 2007 contract. Many exchanges begin each day’s trading with an opening call period for each contract month. The first few bids, offers, and traded prices during this initial time period establish an opening price or an opening range. This can give rise to different open high and open low prices,

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which were $671.20 and $670.90, respectively, for December 2007. Regular trading begins after everyone gets a chance to execute trades at the opening call. High and low are the highest and the lowest traded prices per contract per day. The high price for December 2007 gold was $679.00, while the low was $666.70. The difference between the high and the low prices determines the trading range for the trading session. Like the opening, trading ends in a special way. The closing minutes are usually the busiest because this is when many traders close out their open trades to avoid margins for overnight positions. During the brief ending period, called the market close, or the last minute of trading, known as the closing call, a closing price or a closing range is established. For actively traded contracts with few price fluctuations, the exchange’s Settlement Committee picks a settlement price from this closing range, often the last traded price. The settlement price (abbreviated settle) is the fair value of the contract at market close. For December 2007, the most recent settlement price was $677.50. The task is harder for thinly traded contracts. For example, April 2008 traded nine contracts, which may have traded hours before the close. Moreover, many contracts, like February, April, and June for the year 2009, did not trade at all. In this case, the Settlement Committee considers the spreads relative to other futures prices. The spreads between different maturity futures prices are generally quite stable. For example, the spread between the April and February 2008 settlement prices was $5.70, between June and April $5.60, and between August and June $5.40. Here the spreads between the February and April 2009 and the April and June 2009 settlement prices (contracts with no current trading volume) were set at $5.60. The next column reports the change in the settlement price from the day before. For example, the December 2007 settlement price rose by $0.50 to end at $677.50. Some tables report lifetime highs and lows, which refer to the highest and the lowest traded prices recorded for a contract month since it began trading. This information is absent here. Open interest is the number of outstanding contracts for a particular maturity month. December 2007 has 188,352 open contracts. Each of these contracts must end through a closing trade, a delivery, or a physical exchange. The estimated trading volume during the current session is 59,490 contracts for December 2007, and the last updated gives the time of the last trade.

The Exchange and Clearinghouse US securities exchanges are governed by a tight set of rules and regulations. Traditionally, they have been organized as voluntary, nonprofit institutions. Futures exchanges in the United States have clearinghouses affiliated with them. A clearinghouse forms the foundation on which a futures exchange is built. As noted before, it serves several useful functions: it clears trade (and collects a clearance fee for each contract cleared), it guarantees contract performance to all traders, and it collects margin from

COMMODITY PRICE INDEXES

clearing members. A clearinghouse has matched books; therefore it has no market risk, but it does have counterparty risk. To alleviate this credit risk, it has several layers of financial protection. On top of margins, a member must keep guarantee funds with the clearinghouse. In case a member defaults, the clearinghouse first pays from his margin, then from the member’s guarantee fund, subsequently from the surplus fund of clearance fees, and finally from the guarantee fund of all members according to some prespecified formula.

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8.5

Commodity Price Indexes

We have used equity market indexes, the Dow Jones Industrial Average, and the S&P 500 index to understand the history of stock markets, to gauge market sentiments, to evaluate portfolio performance, and even as the underlying for derivatives. Can indexes also play such roles in commodity markets? The answer is yes. During the 1950s and 1960s, inflation was low, commodity prices were relatively stable, and volatility was not a matter of concern. The 1970s, 1980s, and 1990s exhibited more volatility but no discernable trend in the index. A dramatic upward trend began in 2003 with increased volatility. Not surprisingly, this increased volatility has been accompanied by an increase in the trading of commodity futures. Extension 8.1 presents an interesting debate concerning the direction of future commodity prices between the doomsters and the boomsters. Several commodity price indexes have been created. The most popular commodity price index, the Reuters/Jeffries CRB index (RJ/CRB), is computed by averaging the futures prices of nineteen different commodities. Several other commodity price indexes are popular in the United States. Standard and Poor’s publishes world production–weighted S&P GSCI indexes based on twenty-four commodities, which consist of four separate but related S&P GSCI indexes: the spot index, the excess return index, the total return index, and a futures price index. Introduced in 1998, the Dow Jones–AIG Commodity Index Family (DJ-AIGCI) is composed of indexes based on futures contracts on nineteen physical commodities. DJ-AIGCI indexes are computed on an excess and total return basis, report spot as well as forward indexes, and are available in US dollar and several foreign currency versions. The Rogers International Commodity index (RICI) was developed in 1998 by commodity investor James B. Rogers (see RICI Handbook 2008). Covering more commodities than other indexes, RICI is a dollar-based total return index that corresponds to a collection of commodities representing the global economy including futures traded in different exchanges, in different countries, and quoted in different currencies. Notice the following about these indexes. First, as these indexes are weighted averages, the booming energy sector gets high weights, ranging from 30 plus percent for RJ/CRB and DJ-AIGCI to over 70 percent for the S&P GSCI index in 2008. Second, because these indexes are based on short-lived futures contracts that mature on a regular basis, the managers of the index must periodically roll the futures index

167

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by replacing the expiring futures with similar new contracts. Third, many derivatives have been created based on these indexes. For example, there are exchange-traded funds or exchange-traded notes based on these indexes, and futures or options on futures trade on these indexes. Exchange-traded funds (ETFs) are similar to mutual funds but trade in an exchange on a real-time basis. Exchange-traded notes are bonds issued by an underwriting bank, which promises to pay a return (minus any fees) based on the performance of a market benchmark (like a commodity price index) or some investment strategy.

EXTENSION 8.1: Doomsters and Boomsters

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Two Opposing Views on the Future Direction of Commodity Prices The first camp, popularly known as doomsters, owes its intellectual origins to the great English economist Thomas Malthus, whose ideas on the economics of population growth came to be known as the Malthusian doctrine. It argues that as productivity increases and new lands are discovered, the food supply increases in arithmetic proportions, such as 1, 2, 3, . . . . However, due to the population’s propensity to reproduce, it increases geometrically, such as 1, 2, 4, 8, . . . . Consequently, events like war, famine, and pestilence check the population growth relative to the food supply so that people always stay just at the subsistence level. No wonder economics has been called the dismal science! Building on this intellectual foundation, the doomster school argues that a growing number of humans rely on a limited supply of natural resources, and scarcity will force commodity prices to rise through time. A prominent proponent of this neo-Malthusian view is Stanford University professor Paul R. Ehrlich. His 1968 book The Population Bomb warned, “The battle to feed all of humanity is over. In the 1970s and 1980s hundreds of millions of people will starve to death in spite of any crash programs.” The opposite camp, popularly referred to as boomsters, espouses free-market environmentalism, which argues that free markets, property rights, and a good legal system will take care of the environment. They dismiss the other side’s gloom and doom. They cite that after the high-inflation days of the 1970s, commodity prices failed to show a pronounced upward trend implied by increasing scarcity. The champion of the second group was Julian L. Simon (b. 1932), a professor at the University of Maryland when he died in 1998. His 1981 book The Ultimate Resource challenged the notion of Malthusian catastrophe and offered an alternate explanation. He argued that the power of human beings to invent and adapt is the ultimate resource. Simon showed that after adjusting for inflation and wage increases, most raw material prices fell over the past decades. To quote from a song by the Beatles, “It’s getting better all the time.”

A Bet between Professor Doomster and Professor Boomster Boomster Simon and doomster Ehrlich entered into a wager based on their views (see “When the Boomster Slams the Doomster, Bet on a New Wager—Clash of Eco-Titans Rekindles a Rancorous Core Debate over Future of the Earth,” Wall Street Journal, June 5, 1995). The bet’s conditions were as follows: on September 29, 1980, Ehrlich would buy on paper $200 worth of each of five strategically important metals (chromium, copper, nickel, tin, and tungsten). If, ten years later, their inflation-adjusted prices went up, then Simon would pay the inflation-adjusted value increase for each of these portfolios. If they declined, Simon would receive a payment instead. Note that the professors were trading a derivative that was essentially a combination of five forward contracts with some price adjustments.

SUMMARY

169

Ehrlich and his colleagues saw an arbitrage opportunity: “I and my colleagues, John Holdren (University of California, Berkeley) and John Harte (Lawrence Berkeley Laboratory), jointly accept Simon’s astonishing offer before other greedy people jump in.” Ten years later, they were in for a rude surprise—for a variety of reasons, the real prices for all these metals and the nominal prices for three declined. Wired magazine (“The Doomslayer,” www.wired.com/wired/archive/5.02/ffsimon_pr.html) reports that the drubbing was particularly hurtful because Simon had given Ehrlich and his colleagues a priori advantage by letting them select the five metals. A month after the bet ended, Professor Ehrlich quietly mailed Professor Simon a check for $576.07.

8.6

Summary

1. Both a forward and a futures contract fix a price for a later transaction. They have many differences: unlike a forward, a futures contract is (1) regulated, (2) exchange traded, (3) standardized, (4) liquid, (5) guaranteed by a clearinghouse, (6) margin adjusted, (7) daily settled, (8) usually closed out before maturity, and (9) has a range of delivery dates.

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2. Futures and forwards have many uses: (1) they help smooth out price fluctuations that can come from demand–supply mismatches, (2) they help create a complete market, (3) they help people to speculate, (4) they allow traders to leverage their capital, (5) they make efficient trading possible, and (6) they help generate information and aid in price discovery. 3. Forward trading is as old as antiquity. Today’s futures contract started in nineteenthcentury Chicago and more or less arrived at its present-day form over the next eighty years. An astonishing variety of futures were introduced since the 1970s— financial futures being a major innovation. Moreover, exchanges merged, new exchanges emerged, and electronic trading replaced the traditional open-outcry method of pit trading throughout the world. 4. As futures are standardized, they allow quicker transactions. Futures contract specifications are available on an exchange’s website. 5. Many vendors carry futures price quotes. A typical quote table gives the opening price, the day’s high and low, the settlement price, the change in settlement price from the previous day, lifetime high and low, open interest for each contract, and other information like the estimated volume and open interest for all contracts on that particular commodity. 6. Several commodity price indexes like the Reuters/Jeffries CRB index, S&P GSCI indexes, Dow Jones–AIG Commodity Index Family, and Rogers International Commodity index are used to gauge the direction of commodity prices and to create derivatives that allow investors to add a commodity price exposure to their portfolios.

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8.7

Cases

CME Group (Harvard Business School Case 711005-PDF-ENG). The case describes

the CME Group, the world’s largest commodities exchange, futures and options on futures contracts, history, regulation, and the strategic choices the company faced. Bringing OTC Back to the Exchange: Euronext. liffe’s Launch of ABC

(Harvard Business School Case 706489-PDF-ENG). The case examines value creation, market design, and competitive positioning issues for a derivatives trading exchange in the context of launching matching, clearing, and confirmation services for the over-the-counter market. Lessons Learned? Brooksley Born and the OTC Derivatives Market (A)

(Harvard Business School Case 311044-PDF-ENG). The case studies a proposal to regulate the over-the-counter derivatives market, whose lack of implementation might have been one of the factors that contributed to the financial crisis of 2007–9.

8.8

Questions and Problems

8.1. Where and when was the world’s first modern futures exchange founded?

Explain how trading took place on this exchange. 8.2. Discuss the four different categories into which the history of futures contracts

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since 1970 has been classified, giving two examples of some of the major developments under each of the four categories. 8.3. Consider a fairly illiquid futures contract that has not traded for days. Do you

still need a settlement price for this contract? If so, how would the exchange go about determining this settlement price? 8.4. Explain the difference between closing price and settlement price. Are they

the same or different? Which price is used for marking-to-market? 8.5. If regulation is bad for business, why does the National Futures Association put

a significant amount of regulation on its members? 8.6. Consider the gold futures contract traded in the COMEX division of the CME

group. What is the trading unit size and minimum tick size? 8.7. Using Table 8.5, what is the last price on the June 2008 futures contract? What

does the last price mean? 8.8. Using Table 8.5, what is the open interest on the June 2008 futures contract?

What does the open interest mean? 8.9. Using Table 8.5, looking at the open, high, and low prices on the June futures

contract, what can you tell about the trend of futures prices on this contract during the day? 8.10. Using Table 8.5, which three futures contracts have the most trading activity?

QUESTIONS AND PROBLEMS

8.11. Today, are futures contracts only traded on US exchanges? Explain your answer. 8.12. Is trading in futures contracts mainly for hedging, or speculation, or both?

Explain your answer. 8.13. Explain why forward contracts have been trading for centuries. What economic

function do they perform? What improvement did futures contracts provide over forward contracts? Explain your answer. 8.14. The following questions concern the doomsters and the boomsters. a. What were the opposing views promoted by Professors Ehrlich and Simon? b. Describe the wager between Professors Doomster and Boomster and the

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outcome of this bet.

171

9 Futures Trading 9.1 Introduction 9.2 Brokers, Dealers, and the Futures Industry

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9.3 Trading Futures 9.4 Margin Accounts and Daily Settlement Daily Settlement

9.5 Futures and Forward Price Relations

Equality of Futures and Spot Price at Maturity Equality of Forward and Futures Prices before Maturity Convergence of the Basis

9.6 Trading Spreads 9.7 Summary 9.8 Cases 9.9 Questions and Problems

BROKERS, DEALERS, AND THE FUTURES INDUSTRY

9.1

Introduction

The movie Trading Places made a caricature of futures trading. In a particular sequence, Dan Aykroyd and Eddie Murphy entered the trading pit, visibly nervous and yet pumped up to speculate by shorting orange juice futures. They suddenly realized that the powerful Duke brothers were trying to corner the OJ futures market. This was in anticipation of crop damage from a frost in the orange-growing states. Some traders thought that the Duke brothers had inside information on the extent of the losses and joined the fray. Futures prices increased. Dan, Eddie, and other sellers soon felt that they were sliding down a bottomless pit. Suddenly it all changed. The commerce secretary announced on television that the frost had bypassed the orangegrowing regions. This implied a rich harvest with soft prices. Dan and Eddie were ecstatic: OJ futures prices rapidly fell, and they made a ton of cash; the Duke brothers went completely broke—and all of this happened in an hour! Although the movie is fictionalized, it does capture the dark side of futures trading (it is suspiciously similar to the Hunts brothers’ silver manipulation story in the next chapter). To put this movie in context, this chapter describes the mechanics of trading futures contracts. We discuss brokers, dealers, and others who work in the industry, then we list various ways of moving in and out of futures. This is followed by a description of margins and daily settlement. Next we explore various properties of futures and forward prices. We end with a discussion of futures spread trading strategies.

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9.2

Brokers, Dealers, and the Futures Industry

Brokers match buyers and sellers in futures and earn commissions for this service. They also represent their customers to the exchanges and clearinghouse. Dealers step in and take the other side of trade when no one else offers a better price. Like a used car dealer who maintains an inventory and posts a retail price for trading cars, dealers in futures markets keep an inventory of futures contracts and quote bid and ask prices to maximize income. Moreover, brokers and dealers also act as first-level regulators because they have a duty to report irregularities and fraud to the exchange authorities and government regulators. Brokers and dealers in futures markets may do business as individuals, or they may be organized as associations, partnerships, corporations, or trusts. They must register with the Commodity Futures Trading Commission (CFTC). To open a futures trading account, one must go through a futures commission merchant (FCM) or an introducing broker (IB) registered with an FCM. An IB’s role and responsibilities are limited: an IB may seek and accept orders but must pass them on to a carrying broker (usually an FCM) for execution, and an IB cannot accept funds—funds must be directly deposited with the carrying broker. In contrast, an FCM provides a one-stop service for all aspects of futures trading: solicit trades, take futures orders, accept payments from customers, extend credit to clients, hold margin deposits, document trades, and maintain accounts and trading records. Numerous Associated Persons (APs)

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work for FCMs, IBs, and others, and help people trade (see CFTC’s website https://www.cftc.gov/ConsumerProtection/EducationCenter/registration.html for a description of the intermediaries who facilitate the trading process). Where does one go for trading advice? There are commodity trading advisors who dispense wisdom through newsletters or advise clients on an individual basis. Most of them are technical analysts who use past price patterns to predict future price movements. To invest with professional managers, one can go to commodity pool operators (CPOs). They run commodity pools (funds), which are mutual fund–type operations speculating in futures. Rarely are these pools stellar performers: the returns are often negative, and when positive, they often underperform the market on a risk-adjusted basis. Moreover, they charge large management fees and require big margin deposits.

9.3

Trading Futures

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Entering into a long or a futures position is easy: one calls a broker and places an opening order. But, there are four ways of ending a futures position: a closing transaction, physical delivery, cash settlement, or an exchange for physicals (see Figure 9.1): 1. A closing transaction. Suppose you go long a December gold futures. You are obliged to buy gold at the futures price during the December delivery period. A futures position can be ended earlier. You can do this by an offsetting transaction called a closing transaction or a reversing trade. Just take a short position in the same December gold futures. You are now obliged to sell gold at that day’s futures price during December. This cancels the initial position. Tell your broker that the second trade was a closing transaction, and he will wipe your slate clean. Over 95 percent of futures contracts close before maturity. 2. Physical delivery. This was the only way of closing futures in the past. In a physical delivery, the short gets cash from the long and obtains a warehouse receipt that gives ownership of goods held in a licensed warehouse or a shipping certificate that is a promise by an exchange-approved facility to deliver the commodity under specified terms. A futures contract specifies (1) acceptable deliverable grades (some contracts allow substitution of inferior grades with price discounts and superior grades with price premiums), (2) acceptable delivery dates (usually within the last trading month, i.e., the month that includes the last trading day), and (3) acceptable delivery places (with price adjustments for different locations). Though conceptually similar, the delivery methods vary slightly from exchange to exchange. For most contracts, the short has the privilege of initiating a delivery notice. The exchange usually matches him with the oldest existing long position. In a typical delivery process, there would be a first notice day, a last notice day, and a last trading day. The first notice day is the first day on which a short can submit a notice of intention to make delivery, and the last notice day is the last such day. The last trading day is usually one or more days before the last notice day. Exchanges generally do not make or take delivery of the actual commodity; they only specify how the delivery process occurs.

TRADING FUTURES

FIGURE 9.1: Four Methods of Ending a Futures Position Closing Futures before the Delivery Period Start

Delivery Period

Last Trading Day (a) Closing transaction (preferred way of ending most futures contracts) Ending Futures during the Delivery Period Corn moves Colorado Farmer Corn Processor

Chicago

Kansas City

(b) Physical delivery (inconvenient, large transaction costs, e.g., farmer ships corn to Chicago or Kansas City and delivers corn for cash; processor has to transport corn back to Colorado) (c) Cash settlement (no need to ship the underlying commodity; the contract ends with a final margin adjustment on the last trading day)

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Traders inform exchange

Chicago

Colorado Farmer Corn moves Corn Processor (d) Exchange for physicals (hedgers avoid needless transaction costs, e.g., a farmer and a corn processor arrange delivery for payment and inform the exchange to close positions)

3. Cash settlement. In case of a cash settlement, a final adjustment is made to the margin accounts of the buyer and seller equaling the difference between the last trading day’s settlement price and the settlement price of the previous day—and then the contract ends. 4. Exchange for physicals. Another possibility is a spot market trade between two parties who have genuine buying and selling needs for the underlying commodity and have already hedged their prices by establishing futures positions. This is called an exchange for physicals (EFP) transaction. Afterward, they notify the exchange and the clearinghouse about the transaction, and their slates are rubbed clean. Example 9.1 shows how this works.

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EXAMPLE 9.1: An Exchange for Physicals ■

Suppose a farmer grows corn in Colorado. She sells short fifty corn futures to fix a selling price for her produce. A corn processing plant is located five miles up the road from the farmer. It buys, processes, and sells corn to the cereal maker who makes cornflakes. The corn processor goes long fifty corn futures and fixes a buying price for the input. Both traders are hedgers: the farmer genuinely needs to sell corn, while the processing plant legitimately needs to buy it.

■

At delivery time, why ship 250,000 (= 50 contracts × 5,000 bushels/contract) bushels of corn all the way to Chicago or Kansas City and then ship it back to Colorado as per CME Group’s contract stipulations? This will incur needless transportation costs and time delays.

■

It would be much better to do the transaction directly: the farmer ships corn to the processing plant and receives the day’s settlement price (although provisions allow EFPs at a mutually agreed price). Both report the price and the quantity traded to the Chicago Board of Trade, which then checks that these reports match and closes the fifty long and short positions.

EFPs have grown in popularity in recent years. Most oil futures are settled by EFPs, and their use is increasing in both gold and silver futures markets.

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9.4

Margin Accounts and Daily Settlement

To buy and sell futures, one must open a futures account with a broker. To open a futures position, one must post initial margin (or performance margin), which acts as a performance bond. Most exchanges allow initial margins to be paid with cash, bank letters of credit, or short-term Treasuries. The broker, if she is a clearing member, keeps margin deposits with the exchange’s clearinghouse; if she is not, she keeps them with a clearing member, who in turn keeps margin with the exchange. This earnest money ensures that when the going gets tough, the tough do not leave town!

Daily Settlement When futures prices increase, a long position gains value. This makes sense: if you buy something at a fixed price and the new purchase price increases, then you must make a profit. The seller’s situation is just the opposite. So when the futures price increases, the long’s account is credited with variation margin, which is the change in the settlement price from the previous trading day, and the short’s account is debited the same. These additions (collects) or subtractions ( pays) of variation margins at the end of each trading day are known as daily settlement. When this is done, the margin account is said to be marked-to-market. Traders can keep only a small amount as margin, often 5 percent or less of the position size (e.g., if the gold futures price is $1,000 and the initial margin is $5,000, then it is only 5,000/(1,000 × 100) = 0.05 or 5 percent), and still trade.

MARGIN ACCOUNTS AND DAILY SETTLEMENT

Marking-to-market substantially lowers credit risk and makes futures safer to use than forward contracts. The reason is simple. With futures, the largest loss to the contract is a one-day movement in the futures price. In contrast, with forwards, the largest loss is the movement in the forward price over the entire life of the contract. Margins buffer this smaller loss for futures, whereas collateral buffers this larger loss for forwards. Example 9.2 shows how daily settlement works for a futures contract.

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EXAMPLE 9.2: Daily Settlement of a Gold Futures Contract ■

On Monday, Ms. Longina Long buys three gold futures contracts from Mr. Shorty Short. Both traders have margin accounts. We refer to Monday as time 0, Tuesday as time 1, and so on. The futures contract matures on Friday (time T or 4) when the day’s trading ends. For convenience, we use the end of the day’s spot and settlement prices. All prices and computations are on a per ounce basis. The futures prices are obtained from the cash-and-carry model (see Chapter 11).

■

Monday (time 0) - The spot price on Monday, S(0), is $1,000.00. - The futures price on Monday for the contract maturing on Friday is F(0,4) = $1,000.60. For simplicity, we suppress T from the notation for the futures price F(0,T) and write it as F(0). We will reintroduce it later when considering contracts maturing at different dates. Table 9.1 records the trading day as the first column and futures prices as the second column. - Because futures contracts always clear at fair prices, no cash changes hands when the position is entered at the close of Monday’s trading.

■

Tuesday (time 1) - S(1) = $1,006.00 - F(1) = $1,006.40 - Yesterday, Ms. Long locked in a price of $1,000.60 for buying gold on Friday. Now she is asked to pay $1,006.40 instead. She is understandably upset. However, she will be indifferent if Mr. Short pays her the price difference. By daily settlement, this amount, F(1) – F(0) = 1,006.40 – 1,000.60 = $5.80, is credited to Long’s margin account (and it is debited to Short’s margin account) after Tuesday’s close. This is reported in the third column of Table 9.1. Long’s contract now has a futures price of F(1) = $1,006.40.

■

Wednesday (time 2) - S(2) = $996.00 - F(2) = $996.20 - Long would be happy to buy gold for $996.20 on Friday, while Short would be unhappy to sell at that price. To mark-to-market the contract, Long’s account is debited by a variation margin of F(2) – F(1) = 996.20 – 1,006.40 = – $10.20. Short’s account is credited the same amount. Values become fair again.

■

Thursday (time 3) - S(3) = $988.00 - F(3) = $988.20 - Long’s cash flow is F(3) – F(2) = $988.20 – 996.20 = – $8.

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Friday (time T or time 4) - S(4) = $995.00 = F(4) - The spot and futures prices equal each other at the end of Friday’s trading. They should because a futures contract maturing in an instant is the same as a spot market transaction and therefore the futures price must be the same as the spot price. - Long gets F(4) – F(3) = 995.00 – 988.20 = $6.80 from Short.

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TABLE 9.1: Futures Prices and Margin Account for Longina Long Day

Futures Settlement Price

Change in Futures Settlement Price

Daily Futures Gain or Loss (Last Column × 300)

Mon

F(0) = $1,000.60

Tues

F(1) = 1,006.40

5.80

1,740

Wed

F(2) = 996.20

–10.20

Thurs

F(3) = 988.20

Fri

F(4) = 995.00

Interest Earned on Previous Balance

Margin Account Balance

Margin Call

Margin Account Balance (After Adjustment)

$15,000.00

$15,000.00

1.50

16,741.50

16,741.50

–3,060

1.84

13,683.34

13,683.34

–8.00

–2,400

1.64

11,284.98

6.80

2,040

1.50

17,041.50

3,715.02

15,000.00 17,041.50

How are these adjustments actually made? If your futures position gains value, you may remove funds that are in excess of the initial margin, but if it loses value, the broker gets uncomfortable. So there is a mandatory amount called a maintenance margin (often set at 75 percent of the initial margin) that must be maintained in the account at all times. If the account value touches or drops below this level, then the broker places a margin call and requests that you come up with enough variation margin in cash to bring your account to the initial margin level. If you fail to meet a margin call, your broker can close out your positions. You may be liable for more cash if your position is liquidated at a loss. To avoid this hassle, traders often keep more funds than the required initial margin. In reality, other issues must also be reckoned with. For example, there are transaction costs, and the margin account earns interest. As this discussion can get complicated quickly, to simplify the presentation we embrace the old maxim “follow the money!” Following this maxim, we take an accountant’s perspective and keep track of the margin account’s cash flows, including daily interest (see Example 9.3).

MARGIN ACCOUNTS AND DAILY SETTLEMENT

EXAMPLE 9.3: A Margin Account’s Cash Flows Including Interest Earned ■

Continuing with the previous example, let us track the margin account of Ms. Longina Long, who buys three gold futures contracts on Monday at a futures price of F(0) = $1,000.60 per ounce. We write the number of contracts as n = 3, the contract size as 𝜅 = 100 ounces, and the margin account’s balance as Bal(t), where t = 0, 1, 2, 3, 4 are the days over which the futures contract lasts.

■

As a speculator who is not an exchange member, Long must keep the commodity exchange mandated initial margin of $5,000 per contract or $15,000 in total, which we write as Bal(0). Long’s margin account balances are plotted in Figure 9.2. Her maintenance margin is $4,000 per contract. The margin account balance earns interest. Assume that the interest rate i(0) is 1 basis point for Monday. This overnight interest rate changes randomly across time and gets declared at the start of each trading day.

■

On Tuesday, the gold futures settlement price goes up to F(1) = 1,006.40. Long’s margin account gets credited for the increase in the position’s value and earns interest on the previous day’s balance, values reported in the fourth and fifth columns of Table 9.1, respectively. Tuesday’s margin account balance is

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Bal(1) = [Futures (Tues) − Futures (Mon)] × Number of contracts × Contract size + (1 + Daily interest) × Margin account balance (Mon) = [F (1) − F (0)] × n × 𝜅 + [1 + i (0)] × Bal (0) = 5.80 × 3 × 100 + (1 + 0.0001) × 15, 000 = 1, 740 + 15, 000 + 1.50 = $16, 741.50

■

■

This amount is noted in the sixth column. As there is no margin call, we repeat this in the eighth column. Long can remove the excess margin of $1,741.50 if she wants. We assume that she keeps it in the margin account. On Wednesday, the gold futures price falls to F(2) = $996.20 and Long’s futures position loses value. Assuming Tuesday’s interest rate i(1) is 0.00011 for the day, the new margin balance is Bal (2) = [F (2) − F (1)] × n × 𝜅 + [1 + i (1)] Bal(1) = −10.20 × 300 + 1.00011 × 16, 741.50 = $13, 683.34

■

Her account is now under-margined as it has fallen below the initial margin level. The broker is uncomfortable but does nothing.

■

On Thursday, the gold futures price falls by another $8. Assuming i(2) is 0.00012 for the day, the new margin balance is Bal(3) = $11,284.98. As this is below the maintenance margin level of 3 × 4,000 = $12,000, the broker issues a margin call instructing her to come up with $3,715.02 in cash (which is reported in the sixth column of Table 9.1 and also shown in Figure 9.2). She does this and her

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account balance is restored to the initial margin account level of $15,000, and this figure is reported in the last column. Notice that futures traders are required to bring the margin account balance up to the initial margin level and not to the maintenance margin level as required in stock markets. ■

On Friday, the gold futures price goes up by $6.80. If i(3) = 0.0001, her margin account balance is $17,041.50.

■

Suppose Mr. Shorty Short took the other side of the transaction. His daily futures gains and losses would be the opposite of Long’s, but his margin account balance would earn different interest. Also, as the futures price did not move too much against Short, he will not get any margin calls.

■

As noted in Table 9.1, for the magnitude of these investments, the interest earned is small ($1.50 to $1.84) and perhaps unimportant. However, consider a futures position with a notational in the tens of millions of dollars. In that case, the interest earned per day is considerable and not so easily ignored.

FIGURE 9.2: Long’s Margin Account Balance Margin Account Balance $17,000

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16,000 Initial Margin

15,000 14,000

Margin Call Maintenance Margin

13,000 12,000 11,000 Mon

9.5

Tue

Wed

Thu

Fri

Days

Futures and Forward Price Relations

What is the relation between futures and forward prices? Before addressing that question, let’s consider an example that computes a futures contract’s payoffs when positions may be entered and exited early (see Example 9.4).

FUTURES AND FORWARD PRICE RELATIONS

EXAMPLE 9.4: Futures Payoffs When Positions Are Closed Out Early ■

Consider the data from Example 9.3. Ms. Long bought gold futures on Monday. Her payoff on Tuesday due to daily settlement is Futures (Tues) − Futures (Mon) + One day’s interest on Monday’s margin account balance = F (1) − F (0) + Interest = $5.80 + Interest

■

Instead of on Monday, if Ms. Long buys futures on Tuesday and closes out her position with a reversing trade on Thursday, then her payoff is [Futures (Wed) − Futures (Tues) + One day’s interest on Tuesday’s margin account balance] + [Futures (Thurs) − Futures (Wed) + One day’s interest on Wednesday’s margin account balance] = F (3) − F (1) + Interest = −$18.20 + Interest

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on a per ounce basis. The interest earned on the margin account balance is unknown in advance and hence random.

As indicated in Example 9.4, the interest earnings are dependent on the futures price changes. Furthermore, interest rate changes affect the interest earnings on the margin account as well. In conjunction, these two effects modify the risk of holding a futures contract relative to a forward contract that has no such cash flows. Indeed, if interest rate changes are positively correlated with futures price changes, then the futures position benefits from interest rate movements because as cash flows are received, more interest is earned. In this case, the risk of a futures position is reduced slightly relative to an otherwise identical forward contract position. Conversely, if interest rate changes are negatively correlated with futures price changes, then the futures position suffers from interest rate movements relative to a forward position. Here the risk is increased. It is important to note that this interest rate risk is priced into the market clearing futures prices, driving a wedge between futures and forward prices. Recall that forward contracts have no intermediate cash flows and therefore no explicit interest earnings risk. A generalization of this example gives a useful result for computing a futures trader’s profits and losses. Let date 0 be the initiation date of a futures contract and date T be its delivery date. Suppose a trader enters a long futures position at time t1 (which can be as early as date 0) when the futures price is F(t1 ) and closes out her position at a later date t2 (which can be as late as date T) when the futures price is F(t2 ). A futures trader’s profit or loss is the sum of daily variation margins and interest earned on the margin account balance for the time period over which the futures position is held: Long’s payoff = Closing futures price − Initial futures price + Interest = F (t2 ) − F (t1 ) + Interest

(9.1a)

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and Short’s payoff = − (Closing futures price − Initial futures price) + Interest (9.1b) = − [F (t2 ) − F (t1 )] + Interest on a per ounce basis. Of course, the interest earned on the long and short positions differ. These results make sense and ease the task of computing margin account balances (see Example 9.5). Remember that futures prices are determined in the market so that a trader entering a position pays nothing at the start. If the underlying commodity increases in value, then futures prices also move up. Long benefits because the fixed price at which she agreed to buy has increased in value, and Short loses because what he agreed to sell at a fixed price has become more expensive.

EXAMPLE 9.5: Margin Payments for a Portfolio of Futures Contracts ■

Pearl is a speculator. Trading at the prices given in Table 9.2a, she went long two contracts of gold on January 20, short three platinum contracts on January 21, and long five silver contracts on January 23. She then liquidated all her futures positions on January 26. Her gains and losses are computed in Table 9.2b. Her net gain is Long gold futures payoff + Short platinum futures payoff + Long silver futures payoff = 2, 380 − 1, 785 + 4, 975 = $5, 570

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where we have ignored the interest earned on the margin account balances for simplicity.

Let us continue with Examples 9.2 and 9.3 to explore several important properties of forward and futures prices.

TABLE 9.2A: Futures Prices Trading Date

June Gold Futures, 100 troy oz/Contract (dollars/ounce)

April Platinum Futures, 50 troy oz/Contract (dollars/ounce)

March Silver Futures, 5,000 troy oz/Contract (cents/ounce)

Jan 20

$989.1

$1,149.0

$2,542.1

Jan 21

992.8

1,153.3

2,600.0

Jan 22

989.4

1,154.1

2,553.3

Jan 23

1,007.0

1,169.5

2,623.3

Jan 26

1,001.0

1,165.2

2,643.2

Jan 27

1,008.4

1,175.0

2,651.8

FUTURES AND FORWARD PRICE RELATIONS

TABLE 9.2B: Margin Account for Pearl Portfolio

Computation of Profits/Losses

Long 2 June gold futures on Jan 20

(1,001 – 989.1) per ounce × 100 troy ounce per contract × 2 contracts

$2,380

Short 3 April platinum futures on Jan 21

–(1,165.2 – 1,153.3) × 50 × 3

–1,785

Long 5 March silver on Jan 23

(2,643.2 – 2,623.3) × 5,000 × 5 ÷ 100

4,975

Net profit

Dollar Amount

5,570

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Equality of Futures and Spot Price at Maturity In Example 9.2, suppose that at Friday’s close, the futures price is $995 but the spot price is $997. Then anyone can buy the futures, acquire gold for $995, and sell it immediately in the open market for $997 to pick up an arbitrage profit of $2. The buying pressure raises the futures price, the selling pressure lowers the spot price, and the two converge. If instead the futures price is higher than the spot price, then an arbitrageur can sell futures, make a delivery, and take an arbitrage profit. To prevent arbitrage opportunities, the futures price must equal the spot price at the maturity of the contract. Of course, we need to make assumptions such as that there are no market frictions (such as transaction costs [brokerage commissions, bid/ask spread], margin requirements, short sales restrictions, taxes that may be levied at different rates, and assets that are not perfectly divisible), that spot and futures can be simultaneously traded, and that the futures mature on a single day.

Equality of Forward and Futures Prices before Maturity Are forward and futures prices equal before maturity? They are equal only under a restrictive set of assumptions such as constant interest rates, no market frictions, and no credit risks, but not otherwise. Example 9.6 explores this and related issues.

EXAMPLE 9.6: Equality of Forward and Futures Prices ■

Consider the gold futures contract introduced in Example 9.2. Next, consider a gold forward contract that is similar to this futures contract. It begins on Monday, matures on Friday, and has one hundred ounces of pure gold as the underlying. We compare two trades by Ms. Longina Long entered at Monday’s close: buying one forward contract versus buying one futures contract.

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■

First, assume an idealistic setting in which interest rates are assumed to be zero, market frictions are assumed away, and counterparties have no credit risk. Now let us compare the payoffs on a per ounce basis from forward and futures trades.

■

At Friday’s close, Ms. Long’s payoff from a long forward trade is = Spot (Fri) − Forward (Mon) = 995 − F (Mon)

(9.2a)

where F(Mon) is the forward price for the contract maturing on Friday. ■

At Friday’s close, Ms. Long’s payoff from a long futures trade is = [Futures (Tues) − Futures (Mon)] + [Futures (Wed) − Futures (Tues)] + [Futures (Thurs) − Futures (Wed)] + [Futures (Fri) − Futures (Thurs)] = 995 − F (Mon)

(9.2b)

where F(Mon) is the futures price on Monday. This happens because Futures(Fri) = Spot(Fri) = $995 at Friday’s close. ■

The two payoffs must be equal because they acquire the same asset on the same date while requiring no cash when initiated. Hence, comparing expressions (9.2a) and (9.2b), we get F (Mon) = F (Mon) = $995

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■

(9.3)

Let us allow interest rates to be random, while holding the other assumptions the same. Again, let’s compare payoffs from a forward and futures trade. - At Friday’s close, Ms. Long’s payoff from a long forward trade is = 995 − F (Mon)

(9.4a)

- At Friday’s close, Ms. Long’s payoff from a long futures trade is = 995–F (Mon) + Interest earned on margin account balance over the time period during which futures position is held

(9.4b)

on a per ounce basis. - Equate the payoffs as before and rearrange terms to get F (Mon) = F (Mon) + Interest

(9.5)

■

Expression (9.5) shows that futures and forward prices are structurally different. This is because a futures trader earns interest on her margin account balances but a forward trader does not.

■

Introducing additional market imperfections such as transaction costs serves to complicate the relation between forward and futures prices even more.

TRADING SPREADS

■

In addition, forward and futures contracts face different risks, on top of the market risks discussed earlier. Let us consider the three additional risks stated in Chapter 1 (credit, liquidity, operational) and see how each affects forward and futures contracts differently. - Being exchange traded, a futures contract is essentially free from credit risk; a forward contract may not be. It depends on the collateral relationships prearranged between the relevant counterparties. - Over-the-counter traded forwards are more illiquid than are the exchange-traded futures, introducing greater liquidity risk in forward contracts. - Being an over-the-counter instrument, a forward also has more operational risk than does an exchange-traded futures contract.

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This example demonstrates that under the assumption of zero interest rates, forward and futures prices are equal. But what happens if interest rates are nonzero? In this case, it can be shown (see Jarrow and Turnbull [2000] for a proof) that forward and futures prices are equal, but only if interest rates are nonrandom. This is an improvement over the first assumption, but it is still an unreasonable one. Interest rates are random and wiggle unpredictably across time. If one allows interest rates to be random, which is the truth, then the equality between forward and futures prices no longer holds. The primary reason is that futures earn interest on the margin account, while forward contracts do not. The equality argument that gave us expression (9.3) does not apply. This is demonstrated by continuing the previous example. The moral of the story is that forward and futures contracts are different securities and, consequently, have different prices. They are fraternal twins, not identical!

Convergence of the Basis The spot price minus the futures price is called the basis. Although it randomly changes across time, it eventually becomes zero at maturity. This well-known property of futures prices is known as convergence of the basis. We can easily see from Example 9.4 that the basis takes on the values –0.60 (Monday), –0.40 (Tuesday), –0.20 (Wednesday), –0.20 (Thursday), and 0 (Friday). Figure 9.3a plots the spot and futures prices from Examples 9.2 and 9.3, and Figure 9.3b shows the behavior of the basis over time.

9.6

Trading Spreads

Futures traders trade naked futures or establish various hedge positions. They also trade spreads whereby they try to exploit relative movements between two futures prices that usually track one another for economic, historical, and other reasons. One can set up an intracommodity spread in the futures market by simultaneously buying and selling futures contracts on the same commodity but with different maturity months or an intercommodity spread by simultaneously buying and selling futures contracts on different commodities with the same or different delivery months. When delivery dates differ, it is called a time spread or a calendar spread.

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Why trade spreads? If you are placing directional bets with futures, you need to have expectations about future spot prices. Alternatively, market forces sometimes create abnormal differences between two futures prices that should closely track one another. People speculate with spreads in such situations, hoping to profit when the price difference corrects itself. Notice that when trading spreads, there is no risk from the direction in which the futures prices move. Spread trading can be tricky because spreads may vary with the seasons. For example, although since 1970 wheat has on average cost a dollar more per bushel than corn, an article titled “Corn, Wheat Swap Roles as Prices Surge” (Wall Street Journal, August 9, 2011) noted the opposite: corn was trading higher than wheat, a gap that grew to 66.75 cents in July 2011. This reversal was caused by low supply as well as rising demand for corn from China, while wheat prices fell as many countries had bumper crops. If futures prices move together with spot prices, this is an ideal setting for an intercommodity spread trade that sells corn and buys wheat futures. This strategy was adopted by many traders at the time hoping for “wheat gaining on corn, since the main wheat harvest is over and more corn supplies are poised to hit the market.” But many traders also avoided this strategy in 2011 because, as observed an agricultural consultant, “the seasonal reliability is not there.” Still, spreads are usually safer than an outright long or short position. For this reason, brokers tend to require less margin for spread positions than for naked positions. When closing out a futures or an option position, they regularly ask, Is this part of a spread position? The next two examples illustrate hypothetical spread trading when the futures prices are narrower (see Example 9.7) or wider (see Example 9.8) than expected.

FIGURE 9.3A: Spot and Futures Prices Dollars 1,010 1,005 Spot Prices

1,000

Futures Settlement Prices

995 990 985 980 975

Time 1

2

3

4

5

TRADING SPREADS

FIGURE 9.3B: Behavior of Basis Dollars 0

Time 1

2

3

4

5

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

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EXAMPLE 9.7: An Intracommodity Spread ■

Suppose you notice that the April gold futures price is $1,008, whereas that for December is $1,014 (all prices are per ounce). In traders’ jargon, the spread is $6 to the December side. Sensing a mispricing, you consult your extensive collection of past price data and find that this intracommodity spread has historically hovered around $8.

■

Set up a spread trading strategy: buy the relatively underpriced December futures contract and simultaneously sell the relatively overpriced April futures contract (see Figure 9.4). Traders say that you are buying the spread for $6. No money changes hands because you are trading futures, but you have to keep margin deposits with your broker.

■

Suppose that next month, the April futures price stays the same, but the December futures price rises to $1,016, as predicted. Close out your positions by selling the spread for $8: sell December and buy the April gold futures contract. You make no profit on the April contract but $2 on the December. Stated differently, you make $2 by buying the spread for $6 and selling it for $8.

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Example 9.8 considers an intercommodity spread.

EXAMPLE 9.8: An Intercommodity Spread ■

Suppose that the platinum futures price is $1,080 per ounce and the gold futures price for a contract expiring in the same month is $1,000 per ounce.

■

Suppose you become a technical analyst, put on green goggles, and gaze intently at price charts on the computer screen. You note that the charts reveal that the intercommodity spread should narrow to $50. You can speculate on this insight by selling the spread for $80: sell the relatively overpriced platinum futures and buy the relatively underpriced gold futures contract.

■

Gold futures prices can rise, stay level, or fall, and so can platinum futures prices. Suppose that a month from now, the spread magically narrows to $50. You earn a profit of $30

FIGURE 9.4: Trading Intra-Commodity Spread Today

Next Month Dec futures price $1,016

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December futures price $1,014 Spread $6

Spread $8

April futures price $1,008

April futures price $1,008

Buy Dec and sell April (“buy spread for $6”)

Sell Dec and buy April (“sell spread for $8”)

When trading spreads, remember the wry warning (attributed to renowned economist John Maynard Keynes) that “the market can remain irrational longer than you can remain solvent.”

9.7

Summary

1. One can initiate a futures position by calling a broker and placing an opening order. There are many order placement strategies, the simplest being a market order that is immediately executed. One can move out of a futures position by (1) a closing transaction, (2) physical delivery, (3) cash settlement, or (4) an exchange for physicals. 2. To open a futures position, one must keep an initial margin to guarantee contract performance. When a futures price goes up, a long position gains and a short

QUESTIONS AND PROBLEMS

position loses profits. Through a process called daily settlement, the long’s account is credited with variation margin (which is the change in the settlement price from the previous trading day) and the short’s account is debited the same. The opposite happens when a futures price goes down—long loses and short makes a profit. After daily settlement adjustments are made, the brokerage account is said to be marked-to-market. This substantially lowers credit risk and makes futures safer to use. 3. Futures prices have a number of useful properties. For example, the futures price must equal the spot price at the contract’s maturity. As the contract maturity approaches, the basis (= Spot – Futures price) converges to zero. In an ideal setting (when market frictions are zero), forward and futures prices are equal when interest rates are constant, but unequal when interest rates are random.

9.8

Cases

Investment Linked to Commodity Futures (Harvard Business School Case 2930-

17-PDF-ENG). The case examines an investment linked to an index of commodity futures prices and explores how the index is constructed, how commodity futures behave, and what the portfolio impacts of such an investment might be. Futures on the Mexican Peso (Harvard Business School Case 296004-PDF-ENG).

The case considers the issues that the Chicago Mercantile Exchange faces regarding how to design, and whether and when to introduce, a futures contract on the Mexican peso. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Alcoma: The Strategic Use of Frozen Concentrated Orange Juice Futures

(Harvard Business School Case 595029-PDF-ENG). The case explores price risk management in the orange juice industry when an increase in orange tree production led to a surplus production of orange juice.

9.9

Questions and Problems

9.1. What is a closing transaction on a futures contract? Do you need to take delivery

to terminate a futures contract position? 9.2. What does marking-to-market mean? 9.3. What risk do margin accounts on futures contracts and marking-to-market

minimize? Explain your answer. 9.4. What is the difference between initial margin and variation margin of a futures

contract? Provide a simple example to explain your answer. 9.5. Explain what a margin call is and when it occurs on a futures contract. 9.6. Ignoring credit risk, if interest rates are constant and equal to zero, do forward

prices equal futures prices? Why?

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9.7. What is the basis of a futures contract? Give a simple example to explain your

answer. What does convergence of the basis mean with respect to a futures contract? 9.8. Ignoring credit risk, when interest rates are random, must forward prices and

futures prices be equal? Explain why or why not. 9.9. Suppose you go short one contract of gold at today’s closing futures price

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of $1,300of $1,300 per ounce. Suppose that your brokerage firm requires an initial margin of 5 percent of the position size ($130,000 in this example) and sets the maintenance margin at 80 percent of the initial margin. Contract size is one hundred ounces. Keep the margins constant throughout this example. Track the value of your margin account if the closing futures prices are as follows. Clearly identify any margin call received and the amount of variation margin that you have to produce. Day

Closing Futures Price (dollars/ounce)

0 (today)

$1,300

1

1,303

2

1,297

3

1,290

4

1,297

The next two questions are based on the following data: The euro FX (currency) futures contract that trades in the Merc has the following features:

Euro FX Contract Highlights Ticker symbol

EC

Trading unit (underlying)

Euro 125,000

Quotation

US$ per euro

Minimum price fluctuation (tick)

$.0001/euro = $12.50/contract

Contract months

March, June, September, December

Assume initial margin is $2,511 and maintenance margin is $1,860 per contract (the exchange periodically revises these numbers). 9.10. Track margin account payments to a trader holding long position in two

contracts when the euro takes the following values (in terms of US dollars): 0.8450, 0.8485, 0.8555, 0.8510, 0.8480, 0.8423, 0.8370, 0.8300, 0.8355.

QUESTIONS AND PROBLEMS

9.11. Track the margin account payments to a trader holding short position in one

contract when the euro takes the following values (in terms of US dollars): 0.8450, 0.8485, 0.8555, 0.8510, 0.8480, 0.8423, 0.8370, 0.8300, 0.8355. 9.12. Suppose that the platinum futures price is $1,580 per ounce and the gold futures

price for a contract expiring in the same month as platinum is $1,500 per ounce (see Example 9.8). Hoping that the spread will narrow to $50 in a month’s time, set up a spread trading strategy, discuss all possible outcomes for the futures prices after a month, and illustrate these outcomes in a diagram. 9.13. Explain the difference between cash-settled and physical-settled futures

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contracts.

191

10 Futures Regulations 10.1 Introduction 10.2 Which Markets Have Futures? Price Discovery

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Hedging Speculation EXTENSION 10.1 When Sellers Have Market Power

10.3 Regulation of US Futures Markets Major Regulatory Acts The CFTC, the NFA, and the Regulatory Role of Exchanges

Enforcement Actions by the CFTC and the NFA

10.4 Manipulation in Futures Markets The Hunts Silver Case Salomon Brothers and US Treasury Securities Banging the Close

10.5 Managing Commodity Markets 10.6 Summary 10.7 Cases 10.8 Questions and Problems

WHICH MARKETS HAVE FUTURES?

10.1

Introduction

Price discovery is an important function of futures markets. Even so, US citizens became upset in July 2003 when the Defense Advanced Research Projects Agency (DARPA), a research wing of the US Department of Defense, announced plans for a “Policy Analysis Market,” where anonymous individuals could trade futures whose payoffs would depend on a terrorist indicator. The proposal that the US government set up an online betting parlor for terrorist activities enraged both legislators and the public. The project was killed soon after it was unveiled. Despite its ethical problems, it was hailed as a good idea by some economists, who noted that futures prices have done an excellent job of predicting conventional as well as nonconventional events. DARPA argued that futures markets usually fare better than experts in predicting events like elections, and this project’s mission was to tap the collective wisdom in market prices—price discovery—to understand the probabilities of terrorist activities. But this argument from the Department of Defense didn’t quite fly. This chapter discusses the price discovery role of futures markets, regulations, and market manipulation. First, we explore the market conditions needed for futures to trade. Next, we look at futures market regulations. Finally, we discuss futures trading abuses and market manipulation: colorful stories of powerful players who sought quick gains by cornering the market and squeezing the shorts until regulators foiled their ploys.

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10.2

Which Markets Have Futures?

When is a futures market for a particular commodity likely to succeed? Three conditions—competitive markets, standardization, and volatile prices—seem to be among the key ingredients needed. A competitive market is one in which each investor’s trades are too insignificant to influence prices and traders act as price takers (see Varian 2003). Economists view competitive markets as the ideal setting. Markets cease to be competitive when some participants gain market power and manipulate prices to their benefit. When markets can be manipulated, traders are reluctant to trade knowing that they can lose significant wealth to the manipulators. For example, consider the following: ■

When a monopoly (a single seller), an oligopoly (a small band of sellers), or a cartel fixes commodity prices, futures trading may not develop. This is the case for diamonds but not for oil. Extension 10.1 tells a tale of the spectacularly successful diamond cartel.

■

When a sole buyer or a small number of buyers drive the market, futures markets for the commodity are unlikely. There are no futures markets for missiles and weapons where the government is the sole buyer.

Standardization refers to the ability of a commodity to be sold in standardized units. For example, gold and silver are easy to standardize in terms of quantity

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and quality. As such, one can easily write contracts on future purchases or sales of these commodities in large quantities. In contrast, automobiles are not so easily standardized. There are hundreds of different car models available, distinguished by manufacturer and other features like manual or automatic, electric, gas, or diesel, and so on. Consequently, it is no surprise that automobile futures do not exist. Volatile prices are commodity prices that change randomly across time in an erratic (up and down) fashion. Volatile prices create risk for producers in determining both their input costs and output sales. Volatile prices therefore generate a demand to hedge these price risks and also a need to forecast future prices. Futures markets exist to fulfill these two needs. Let us explore both these roles filled by futures markets, price discovery, and hedging in more detail.

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Price Discovery Price discovery is related to the notion of market efficiency (see Chapter 6). In efficient markets, prices accurately reflect available information. Such information includes past and current prices (weak-form efficiency), publicly available information (semi-strong-form efficiency), and private information (strong-form efficiency). This concept is not only true for spot markets but for futures markets as well. Because futures market traders include producers who have private knowledge of supply and demand conditions, futures markets are widely believed to be (semistrong-form) efficient, thereby generating prices that reflect more information and provide better price forecasts—price discovery—than any ordinary trader’s information could. This is the motivation underlying the US Department of Defense’s July 2003 proposal to set up a futures market in terrorism indicators. Although well intentioned, the plan was shelved following heavy criticism concerning its ethical considerations. Price discovery helps futures buyers and sellers of commodities. For example, firms that process and distribute corn can fix the buying and/or selling price by trading futures, or the retail seller, which may be your neighborhood supermarket chain, can do the same.

Hedging Hedging in the New Oxford Dictionary of English is defined as to “protect (one’s investment or an investor) against loss by making balancing or compensating contracts or transactions.” If you hedge with futures (for which you pay nothing to start), then you get protection from price swings. You benefit when the spot adversely moves but surrender gains when the spot moves in your favor. Hedging with futures is beneficial to firms for minimizing input and output price risks, thereby smoothing costs and profits. This is true for both producers (e.g., farmers, manufacturers) and suppliers (e.g., gold and copper mines). Hedging is the motivation for the existence of oil futures markets, despite that the oil producers form a pricing cartel known as the Organization of the Petroleum Exporting Countries (OPEC).

WHICH MARKETS HAVE FUTURES?

195

Speculation In contrast to hedging, the term speculation carries a negative connotation. The humorist Mark Twain said, “There are two times in a man’s life when he should not speculate: when he can’t afford it, and when he can.” But speculation is really the flip side of hedging. Hedgers are buying insurance against adverse price movements. Speculators are selling the insurance: gambling that the adverse price movements will not occur. For futures markets to work, both hedgers and speculators are usually needed, and both play a crucial role. The reason is that it is unusual for hedgers on both sides of the market (suppliers and users) to have exactly offsetting demands. As such, hedging demands for a commodity are often one-sided. In such circumstances, speculators provide the liquidity needed for a successful futures market to exist. In addition to their role in providing market liquidity—being the insurance sellers—speculators also play a second important role in futures markets: they facilitate the price discovery role because they often trade based on private information, obtained by costly investigation. Speculators trading on the basis of their private information increase market efficiency because as they trade, their information gets reflected in market prices. Without speculators, futures markets would be both less liquid and less efficient, and sometimes, without speculators, futures markets would not exist. An example is the market for precious diamonds, in which the Diamond Cartel removed the speculators’ incentives to participate (see Extension 10.1).

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EXTENSION 10.1: When Sellers Have Market Power

Cartels An oligopoly is a market that is dominated by a handful of sellers, more than one but not too many. For example, Boeing of the United States, Airbus of Europe, and some fringe suppliers like Ilyushin and Tupolev (now divisions of Russia’s United Aircraft Corporation) dominate the production of large passenger airplanes. Members of an oligopoly sometimes join to form a cartel. A cartel is a coalition of producers that monopolizes a commodity’s production and sale. A cartel tries to maximize joint profits by controlling supply—it allocates a share of the common market to each member, sets sales quotas, regulates production, and fixes prices. By eliminating competition, a cartel raises prices for consumers and profits for the producers. In the process, it may keep inefficient companies in business. Cartels are banned in the United Kingdom and the United States. However, they are legal in many parts of the world, including highly industrialized nations like France, Germany, Italy, and Japan. By controlling the supply of a commodity, and therefore prices, cartels can potentially eliminate the benefits of both hedging and price discovery in futures markets. In such circumstances, futures markets will not exist. This is the case with the Diamond Cartel.

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De Beers and the Diamond Cartel

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Cecil Rhodes founded this enterprise during the 1880s by consolidating several diamond mines in Southern Africa. Around the same time, he formed the London Diamond Syndicate to control the world’s diamond trade. This was later transformed by Sir Ernest Oppenheimer into the Central Selling Organization, a group of marketing organizations that controlled much of the world’s diamond trade until the last decade of the twentieth century. A February 2001 Fortune magazine article (“The De Beers Story: A New Cut on an Old Monopoly”) reported that De Beers entered agreements with most major diamond producers (including the former Soviet Union) and led a very effective cartel. In the 1990s, about 45 percent of the global diamond production was mined by De Beers and another 25 percent went through its hands. In 2000, De Beers announced that it would stop manipulating the supply by buying and storing diamonds. Afterward, De Beers focused on selling diamonds from its own mines after branding them with a “Forevermark” to guarantee their integrity. These steps strengthened the demand for diamonds à la De Beers. Despite this announcement, the effectiveness of the diamond cartel did not disappear. A New York Times article dated August 9, 2008, titled “Talking Business: Diamonds Are Forever in Botswana,” reported that “today, De Beers has about 40 percent of the diamond market—but it is far more profitable than under the old regime, when it controlled 80 percent of the market.” The strong control of the supply of diamonds by the De Beers Group destroyed the price discovery role of a diamond futures market as documented because diamond prices do not reflect supply and demand but the activities of De Beers. Although hedging demands certainly exist for firms using diamonds in production, there are no offsetting hedging demands from the suppliers (the cartel). For a diamond futures market to work, speculators need to provide the missing liquidity, but the strong control of the market supply by De Beers removed the desire for speculators to participate in such a market. Consequently, no diamond futures markets exist today.

10.3

Regulation of US Futures Markets

Futures markets are not for the fainthearted. Wild price swings have been common since inception, and there have been numerous attempts to manipulate futures prices. Competitive markets can lead to economic growth as well as efficiency. However, sometimes they fail to function, creating a need for regulation. As is often the case with regulations, many of the rules and regulations of the US securities market have arisen to fix problems evidenced within financial markets during a financial crisis.

Major Regulatory Acts Regulation of the US futures markets began on a cold winter’s day in February 1859, when the governor of Illinois signed a legislative act that granted a corporate charter to the Chicago Board of Trade (CBOT). The charter, among other things, granted the exchange self-regulatory authority over its members, standardized commodity grades,

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REGULATION OF US FUTURES MARKETS

and provided for CBOT-appointed grain inspectors whose decisions were binding on members. This was a critical early step in the evolution of futures contracts.1 Another major step was taken in October 1865, when formal trading rules were introduced at the CBOT, including margin requirements and delivery procedures. Three years later, the exchange adopted a rule to deter manipulation by banning “‘corners’ (defined as ‘making contracts for the purchase of a commodity, and then taking measures to render it impossible for the seller to fill his contract, for the purpose of extorting money from him’).”2 In 1877, the CBOT began publishing futures prices on a regular basis, and in 1883, the first organization was formed to clear CBOT contracts on a voluntary basis. US federal regulations with respect to futures markets began in the 1880s when the first bills “to regulate, ban, or tax futures trading in the U.S.”3 were introduced in Congress. About two hundred such bills were introduced over the next forty years (see Table 10.1 for a list of some major US futures regulations). As agricultural futures dominated the markets in the early days, regulations primarily focused on grain (such as corn, wheat, oats, and rye) trading, and the regulatory controls were placed in the hands of the US Department of Agriculture (USDA). In 1922, the Grain Futures Act was passed. It regulated grain futures trading, banned “off-contract-market futures” trading, created (an agency of the USDA) the Grain Futures Administration for administering the act, and established the Grain Futures Commission (consisting of the secretaries of agriculture and commerce and the attorney general), which had the broad power of suspending or revoking a contract market’s designation. The Commodity Exchange Act of 1936 amended and extended the 1922 act. It replaced references to “grains” with the term “commodities,” and it expanded the approved list to include commodities like butter, cotton, and eggs. The Grain Futures Commission became the Commodity Exchange Commission and was granted the authority to establish federal speculative position limits. In 1947, this commission evolved into the Commodity Exchange Authority (also an agency of the USDA). It continued to administer the Commodity Exchange Act until the mid-1970s. But all this changed with the passing of the Commodity Futures Trading Commission (CFTC) Act of 1974. This created the CFTC as the independent, ultimate “federal regulatory agency for futures trading,” taking responsibility for regulating futures trading in 1975. The CFTC Act also proposed creation of a self-regulatory organization. Subsequently, an industry body called the National Futures Association (NFA) was established in 1982, which works with the CFTC to help regulate the markets. More legislation followed in 1978 and 1982. This whole body of legislation is known as the amended Commodity Exchange Act (CEA).

1

See “History of the CFTC: US Futures Trading and Regulation Before the Creation of the CFTC” from www.cftc.gov/About/HistoryoftheCFTC/history_precftc. 2

ibid.

3

ibid.

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TABLE 10.1: Some Major Milestones in the Federal Regulation of US Futures Markets Year

Development

1880s

The first bills are introduced in Congress to regulate futures markets.

1922

The Grain Futures Act was passed. The Grain Futures Administration was created as a department of the US Department of Agriculture.

1936

The Commodity Exchange Act, an amendment of the 1922 Act, was passed. The Commodity Exchange Administration was created as the first federal regulator of futures markets.

1974

The Commodity Futures Trading Commission Act was passed. This created the Commodity Futures Trading Commission as an independent “federal regulatory agency for futures trading.”

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1978, 1982

The Futures Trading Acts were passed.

1982

The National Futures Association, a self-regulatory agency for the futures industry, was established.

2000

The Commodity Futures Modernization Act of 2000 was passed.

2010

The Dodd–Frank Wall Street Reform and Consumer Protection Act was passed. This significantly increased regulatory oversight of the over-the-counter derivatives market.

A landmark legislation that substantially altered the CEA, the Commodity Futures Modernization Act of 2000 (CFMA), was passed by Congress and signed into a law in December 2000. The major changes included the following: 1. Lifting of a long-standing ban by the CFMA, which made it possible to trade security futures products, which are futures contracts based on single securities and narrowly based stock indexes. 2. Easing up on regulations for swaps, other over-the-counter (OTC) derivatives, and trading facilities for these products. 3. Streamlining of regulation and the creation of three types of markets: (1) designated contract markets (DCMs, which are boards of trade [or exchanges] that operate under the regulatory oversight of the CFTC), (2) derivative transaction execution facilities, which have fewer regulations but more restrictions on who and what trades, and (3) exempt boards of trade with restrictions similar to (2) but exempt from CFTC regulation, except for antifraud and antimanipulation provisions. The CFMA regulations reflected the deregulatory mood prevalent at the time. The loosely regulated OTC derivative markets seemingly worked well until the financial

REGULATION OF US FUTURES MARKETS

crisis of 2007–9, where it was documented that trading in OTC derivatives helped cause the crisis. In response to this lack of regulation, in 2010, Congress passed the Dodd–Frank Wall Street Reform and Consumer Protection Act. This massive 849-page act4 was the largest effort to regulate US financial markets since the 1930s Great Depression. It introduced numerous significant changes to bank and financial market regulation. Some key features of the Dodd–Frank Act related to derivatives are as follows:5 ■

Increased regulation. Provides the SEC and CFTC authority to regulate OTC derivatives so that irresponsible practices and excessive risk taking no longer escape regulatory oversight.

■

Central clearing and exchange trading. To require central clearing and exchange trading for more derivatives, this act replaced derivatives transaction execution facilities with a new type of entity, the swap execution facility.

■

Market transparency. This requires data collection by clearinghouses and swap repositories to improve market transparency and provide regulators the necessary information for monitoring and responding to systemic risks.

■

Financial safeguards. These add safeguards by ensuring that dealers and major swap participants have adequate financial resources to guarantee the execution of derivatives contracts.

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The CFTC, the NFA, and the Regulatory Role of Exchanges In establishing the CFTC, Congress recognized that futures markets are important to economic growth. The CFTC carries out a variety of activities (see https://www.cftc.gov/About/CFTCOrganization/index.htm): ■

The CFTC must approve new contracts and changes to existing contracts. Each new futures contract is evaluated with respect to three issues: (1) justification of individual terms and conditions, which thoroughly examines the details of the contract to ensure that there will be sufficient supply of the underlying commodity to prevent market manipulation; (2) an economic purpose test, which requires exchanges to show that the proposed contract will be useful for price discovery and hedging; and (3) other public interest requirements.

■

The CFTC develops rules and regulations that govern the NFA and all futures exchanges. These rules and regulations define the requirements for registration, disclosure, minimum financial standards, daily settlement, separation of customer funds, supervision and internal controls, and other activities.

4 5

See www.gpo.gov/fdsys/pkg/PLAW-111publ203/pdf/PLAW-111publ203.pdf.

See http://banking.senate.gov/public/_files/070110_Dodd_Frank_Wall_Street_Reform_comprehensive_summary_Final.pdf.

199

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■

The CFTC requires companies and individuals who handle customer funds or give trading advice to register with them. Actually, the NFA handles the registration process on the CFTC’s behalf.

■

The CFTC ensures compliance of its rules and regulations as well as the Commodity Exchange and Dodd–Frank acts. It conducts trade practice surveillance and audits selected registrants. It investigates and prosecutes alleged violations of CFTC regulations. Whenever necessary, the CFTC files court cases.

■

The CFTC detects and prevents manipulation, congestion, and price distortions. It conducts daily market surveillance. It can intervene and take corrective action if it believes manipulation is present.

■

The CFTC oversees training of brokers and their representatives and other industry professionals. It can force brokers and representatives to take competency tests. The CFTC also makes sure that all registrants complete ethics training.

■

The CFTC does research on futures markets and provides technical assistance. It makes economic analyses for enforcement investigations and gives expert help and technical aid with case development and trials to US attorney’s offices, other federal and state regulators, and international authorities.

■

The CFTC helps coordinate global regulatory efforts. It develops policies and regulations governing foreign and cross-border transactions.

■

The CFTC handles customer complaints against registrants. It hears and decides enforcement cases. The CFTC offers a reparations procedure for persons who have reason to believe that they have suffered a loss due to a violation of the Commodity Exchange Act or CFTC regulations in their dealings with a CFTC registrant.

The SEC regulates securities and the CFTC regulates futures, but who regulates options? Following an intense turf battle between these two regulators, the agreement is that the SEC regulates options on stocks, while the CFTC regulates those on futures. The fight relapses whenever new derivatives are introduced that carry joint characteristics. The issue of overlapping jurisdiction remains unresolved at this time. On the next rung of the regulatory ladder are the commodity exchanges and the National Futures Association. These self-regulatory organizations (SROs) must, among other things outlined in the CFTC’s regulations, enforce minimum financial and reporting requirements for their members. As a registered futures association under the Commodity Exchange Act, the NFA is an SRO composed of futures commission merchants, commodity pool operators, commodity trading advisors, introducing brokers, leverage transaction merchants, commodity exchanges, commercial firms, and others who work in the futures industry.6 Banks and exchanges may join the NFA, but they are not required to do so. The NFA is involved in a range of activities that include the following:

6

www.nfa.futures.org.

REGULATION OF US FUTURES MARKETS

■

On behalf of the CFTC, the NFA registers all categories of persons and firms dealing with futures customers.

■

The NFA screens and tests registration applicants and determines their qualifications and proficiency.

■

The NFA requires futures commission merchants (FCMs) and introducing brokers to keep sufficient capital and maintain good trading records.

■

The NFA tracks financial conditions, retail sales practices, and the business conduct of futures professionals, and it ensures compliance with the requirements.

■

The NFA audits, examines, and conducts financial surveillance to enforce compliance by members.

■

The NFA maintains an arbitration program that helps to resolve trade disputes.

Commodity exchanges complement federal regulations with their own rules. These rules cover many aspects of futures trading such as the clearance of trades, trade orders and records, position limits, price limits, disciplinary actions, floor trading practices, and standards of business conduct. Clearinghouses also take part in regulatory activities.

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Enforcement Actions by the CFTC and the NFA The NFA has comprehensive compliance rules “covering a wide variety of areas such as advertising, telephone solicitations, risk disclosure, discretionary trading, disclosure of fees, minimum capital requirements, reporting and proficiency testing.” The NFA takes disciplinary action against violators, ranging from issuing warning letters to filing formal complaints that can result in penalties that include “expulsion, suspension for a fixed period, prohibition from future association with any NFA Member, censure, reprimand and a fine of up to $250,000 per violation.” The CFTC’s Division of Enforcement investigates and prosecutes alleged violations of the Commodity Exchange Act and Commission regulations. The CFTC takes enforcement action against individuals and firms registered with the commission, those who are engaged in commodity futures and option trading on designated domestic exchanges, and those who improperly market futures and options contracts. The CFTC also vigorously investigates and prosecutes Ponzi schemes. Named after the notorious swindler Charles Ponzi, a Ponzi scheme is a scam that pays early investors returns from the cash coming in from subsequent investors. A common technique is to start a commodity pool and lure unwary, unsophisticated investors by promising them a generous return. Eventually, the whole scheme unravels and leaves behind a trail of cheated customers. Ponzi schemes have attracted special attention since 2008, when it was revealed that prominent stockbroker and financial adviser Bernard Madoff ran a giant Ponzi scheme that bilked investors for tens of billions of dollars.

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10.4

Manipulation in Futures Markets

In 1869, notorious financiers Jay Gould, James Fisk, and their co-conspirators tried to corner the gold market. As told in “Gold at 160. Gold at 130” (Harper’s Weekly, October 16, 1869), the conspirators bought up huge amounts of gold and attempted a market corner. The result was a two-week period of turmoil in the gold market that brought America’s foreign trade, which used gold as the medium of exchange, to a virtual standstill. To defuse a looming economic panic and restore normal conditions in the gold market, President General Ulysses Grant ordered the sale of gold to buy bonds. On September 24, 1869, minutes after the Treasury secretary’s telegram instructing a sale of $4 million worth of gold reached the “Gold Room,” the price of gold, which had earlier ranged between $160 and $162, fell to $133 per ounce. The day went down in history as Black Friday, and the speculators got a severe jolt. This was just one incident. As we noted in Chapter 6, market corners and short squeezes have a long history in financial markets. Market structure, the creation of shorts as futures get transacted, and supply variability for many of the underlying commodities make futures markets especially prone to price manipulation and trading abuses. We now discuss some well-known cases of manipulation in these markets.

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The Hunts Silver Case A Peruvian government firm Minpeco SA sued the Hunt brothers and “coconspirators (including Arab sheiks)” for $150 million, charging that they conspired to corner the silver market. A jury found them guilty, and the Hunt brothers were given huge fines. A class-action suit was also filed on behalf of seventeen thousand investors. Eventually, in the late 1980s, two of the Hunt brothers filed for bankruptcy. The details were as follows (see Duffie [1989], pp. 335–9 and Edwards and Ma [1992], p. 198–9): ■

Oil magnate H. L. Hunt, one of world’s richest persons, had two wives. One of his wife’s sons (referred to here as the Hunt brothers or the Hunts) had a mindboggling net worth of over $5 billion in 1980 but ended bankrupt after the Silver Crisis.

■

Several times during the 1970s, the Hunts took large long positions in both silver spot and futures markets and rolled their hedge forward (i.e., closed out a position in an expiring futures and then took an identical position in a distant-month futures contract). They also invested in a number of silver producers.

■

This pattern was repeated in 1979–80 on a grander scale. The Hunt brothers (1) held large long positions in silver futures (at times reaching nearly 250 million ounces—an amount equivalent to fifty thousand COMEX contracts [that traded in the Commodity Exchange] and much more than US domestic silver consumption), (2) held dominant long positions in many near-month contracts, (3) demanded delivery when contracts matured, and (4) made supply scarce by holding huge quantities of silver bullions and coins. Some twenty futures commission merchants helped the Hunt brothers, and the firms Bache Halsey Stuart Shields and Merrill Lynch handled over 80 percent of the Hunt brothers’ spot and futures trading.

MANIPULATION IN FUTURES MARKETS

■

Silver prices shot up. Historically, silver prices rarely crossed $10 per ounce, but the spot and futures price of silver rose from about $9 per ounce in July 1979 to $35 per ounce at year’s end, peaking at over $50 per ounce in January 1980. Apparently, the Hunts had cornered the market.

■

The exchanges consulted the CFTC and took steps to break the squeeze. Margin requirements were increased, stricter position limits were imposed, and the Hunts were forced to implement liquidation-only trading. They could not increase their silver position unless it was for hedging purposes. They also had to exit the market as contracts expired. By the end of March, silver prices dropped to $11 per ounce. It was this decline in silver prices that destroyed the Hunts’ wealth.

Salomon Brothers and US Treasury Securities

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The US Treasury securities market is one of the world’s largest and most liquid securities markets. Legal scholars like Easterbrook (1986) and Fischel and Ross (1991) have opined that it is nearly impossible to corner the supply of Treasuries. Unfortunately, the impossible occurred as Salomon Brothers Inc., a major player in the Treasury securities market, admitted deliberate and repeated violations of Treasury auction rules beginning in 1990. Here is how the Salomon short squeeze in the May 1991 auction of the two-year US Treasury note worked: ■

There are three related markets for buying Treasuries: (1) the sealed bid Treasury auction, (2) the preauction when-issued market, and (3) the postauction resale market (see Chapter 2).

■

Salomon took considerable pride in its trading prowess and boasted itself to be the greatest bond trading firm in the world. To ensure broad participation (read, stop Salomon’s dominance!) in its auction of Treasury securities, the Treasury enacted the 35 percent rule for each auction: a single bidder cannot bid more than 35 percent of the offering at any particular yield or be awarded more than 35 percent of the offering.

■

Many Wall Street firms sell Treasuries to their clients in the when-issued market, enter the auction with a net short position, and buy securities in the auction to fulfill their obligations. This is a profitable business because the price usually goes down slightly from the when-issued price owing to greater supply in the auction— but not in this auction. Salomon submitted aggressive bids at a yield of 6.81 percent when the notes were trading on a when-issued basis at approximately 6.83 percent directly prior to the auction (which means Salomon paid more for the auctioned securities).

■

Salomon evaded the 35 percent rule by submitting multibillion dollar bids on behalf of clients and then transferred those securities to its own account at cost. Salomon ended up controlling 94 percent of the competitively auctioned securities. Having cornered the market, Salomon squeezed the shorts from the when-issued market by charging them a premium price for these notes when they came to cover their short positions in the post auction resale market.

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■

Academics have tried to measure how much Salomon stood to gain from this squeeze. Studies estimate that this ranged between $5 million and $30 million (for details, see Jegadeesh 1993; Jordan and Jordan 1996).

■

The penalties to Salomon were enormous. Its reputation was badly bruised. Soon after the cornering allegations surfaced and federal agencies started investigating, regulators barred Salomon from making a market in various federal and state government securities, cutting off a major source of the company’s income. Salomon stock, which was trading in the $50 to $60 range, soon plunged to $30. Billionaire Warren Buffett, a major shareholder of Salomon, flew in from Nebraska and took control. Several top executives lost their jobs, and many talented personnel left. Salomon admitted no wrongdoing other than violating Treasury auction rules and paid a staggering $290 million in fines. The once-feared Salomon now belongs to the past. It went through mergers and acquisitions and is now part of Salomon Smith Barney, which is a subsidiary of Citigroup. Even the name Salomon has been phased out.

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Banging the Close Technology has opened up new ways of manipulating prices. On July 24, 2008, the CFTC filed a complaint against Netherlands-based Optiver Holding, two of its subsidiaries, and three Optiver employees in a US district court, charging that the defendants had manipulated or attempted to manipulate NYMEX crude oil, gasoline, and heating oil futures contracts on nineteen separate instances in March 2007 and succeeded in creating artificial prices in at least five of those cases.7 The defendants traded three NYMEX energy futures contracts: the Light Sweet Crude Oil futures contract (Crude Oil, also referred to as West Texas Intermediate [WTI]), the New York Harbor Heating Oil futures contract (Heating Oil), and the New York Harbor Reformulated Gasoline Blendstock futures contract (New York Harbor Gasoline). The CFTC alleged that the defendants employed a practice popularly known as banging (or marking) the close, which involves manipulating the prices by trading a large position leading up to the close followed by offsetting the position before the end of trading. The scheme had the following ingredients: ■

It involved Trading at Settlement (TAS) contracts for crude oil, heating oil, and gasoline contracts. These are special futures contracts in which the counterparties decide at the time of trading that the contract price will be the day’s settlement price plus or minus an agreed differential. If you trade a TAS contract, you can offset it by trading a futures contract on the other side of the market.

■

Computing settlement prices is tricky in many futures markets because of potential manipulation. The settlement prices for each of these three energy futures contracts

7

Our discussion is based on “Complaint: Optiver US, LLC, et al.,” “CFTC Charges Optiver Holding BV, Two Subsidiaries, and High-Ranking Employees with Manipulation of NYMEX Crude Oil, Heating Oil, and Gasoline Futures Contracts” and “Case Background Information re CFTC v. Optiver US, et al.” (www.cftc.gov/PressRoom/PressReleases/ pr5521-08).

MANAGING COMMODITY MARKETS

are determined using the volume-weighted average prices (VWAP) of futures trades occurring during the closing period (the close) for the contracts, which lasts from 2:28 to 2:30 PM. The defendants planned to control the VWAP and wrote about “bullying the market” in their recorded conversations and e-mails. ■

The traders concocted a three-step system to make manipulative profits: - First, they would accumulate substantial positions in TAS contracts by trading at different times of the trading day. - Second, they would enter closing trades for approximately 20 to 30 percent of their contracts during the pre-close period, which are the dying minutes before the market enters the close. - Third, Optiver would sell 70 to 80 percent of its holdings during the close, which lasts from 2:28:00 to 2:29:59. In conjunction, these three steps manipulated prices so that Optiver could obtain profits from its trades.

■

The CFTC alleged that the defendants had forced the futures prices lower on three instances and higher in two cases and made approximately $1 million in the process.

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10.5

Managing Commodity Markets

The economic definition of a speculator is someone who is trading and not hedging. For example, a trader who is short gold futures but does not hold gold in inventory is speculating. In contrast, if this trader holds gold in inventory for production purposes, then she is a hedger. Futures regulations regard speculators as “evil” and impose size limits on their positions—to preclude the accumulation of market power for the purposes of manipulative trading strategies. In contrast, they view hedgers as “saintly” and impose no such size limit restrictions. Although the economic definition of a speculator and hedger is unambiguous, in practice, it is not always so easy to tell speculators from hedgers. Prior to the 1990s, the CFTC and exchanges classified futures traders like farmers, manufacturers, and commodities dealers (with legitimate hedging needs) as commercial traders, and the CFTC allowed them to trade virtually any number of exchange-traded futures contracts desired. These were the hedgers. By contrast, they imposed speculative position limits on all other traders, who were viewed as noncommercial traders. These were the speculators. Obviously, noncommercial traders could be hedgers using the economic definition, but they were not considered as such. However, beginning in the 1990s, two new kinds of traders, commodity index traders and swap dealers, entered the markets and changed these traditional classifications. Commodity index traders include institutional players like endowment funds and pension funds. They diversify their portfolios by adding commodity exposures through passive long-term investment in commodity indexes. Swap dealers play a more complex role—on one hand, they compete with the futures markets by offering customized derivatives to their clients, but on the other hand, they

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also trade in the futures markets to hedge risks in their swap books. Typically affiliated with a bank or other large financial institution, swap dealers operate as market makers by being ready to act as the counterparty to both commercial and noncommercial traders. A change in the regulatory framework took place in 1991, when the CFTC granted a Goldman Sachs subsidiary J. Aron the same exemption from speculative position limits as those allowed to commercial traders (see “A Few Speculators Dominate Vast Market for Oil Trading,” Washington Post, August 21, 2008). J. Aron was into commodity merchandising and traded swaps as a part of its business. It planned to sell to a large pension fund a commodity swap based on an index that included wheat, corn, and soybeans, all of which fell under federal speculative position limits. To hedge this short commodities exposure, it planned to buy exchangetraded futures contracts on those commodities. The CFTC classified this as a bona fide hedge because the swap dealer had demonstrated that the positions were “economically appropriate to the reduction of risk exposure attendant to the conduct and management of a commercial enterprise.” Once the door was opened, more exemptions followed.

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10.6

Summary

1. Futures contract trading in the United States is a highly regulated activity. It is governed by the Commodity Exchange Act. The Commodity Futures Modernization Act of 2000 made substantial changes to the existing legislation. It made it possible to trade security futures products, it eased up on regulations of OTC derivatives, it allowed new types of futures trading facilities and products, and it facilitated clearing of OTC derivative trades. In 2010, the Dodd-Frank Wall Street Reform and Consumer Protection Act was passed to increase OTC derivative market regulation in response to the financial crisis of 2007–2009. 2. Market regulation in the United States is implemented at the highest level by the federal regulatory agency, the Commodity Futures Trading Commission, and then by a self-regulatory organization, the National Futures Association, the exchanges themselves, and finally, at the grassroots level, by brokers handling customer accounts. 3. The Commodity Futures Trading Commission monitors the markets, investigates and prosecutes alleged violations of the Commodity Exchange Act and commission regulations, and punishes violators with fines and (temporary or permanent) banishment from the futures industry.

10.7

Cases

Amaranth Advisors: Burning Six Billion in Thirty Days (Richard Ivey School of

Business Foundation Case 908N03-PDF-ENG, Harvard Business Publishing). The case provides students with (1) a deeper understanding of commodity futures markets in general and of natural gas markets in particular, (2) an introduction to

QUESTIONS AND PROBLEMS

hedge funds and an insight into the largest hedge fund collapse in history, and (3) an introduction to such concepts as liquidity risk, value-at-risk, spread trades, and the use of derivatives. Mylan Lab’s Proposed Merger with King Pharmaceutical (Harvard Business

School Case 209097-PDF-ENG). The case considers how hedge funds and other investors may use derivatives to separate votes from shares and the legal, moral, and economic implications of this ability. Rogue Trader at Daiwa Bank (A): Management Responsibility under Different Jurisprudential Systems, Practices, and Cultures (University of Hong Kong

Case HKU442-PDF-ENG, Harvard Business Publishing). The case examines the importance of complying with regulations in the context of a major foreign bank operating in the United States.

10.8

Questions and Problems

10.1. Give two reasons why futures markets may not be started on a commodity.

Give an example of such a commodity to support your answer. 10.2. What is price discovery with respect to the trading of a futures contract? Give

an example of its use. 10.3. Explain why hedging and speculation are like “two sides of the same coin.” 10.4. What is the CFTC’s mission? 10.5. What is the National Futures Association and how does it differ from the Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

CFTC? 10.6. If regulation is bad for business, why does the National Futures Association

put a significant amount of regulation on its members? 10.7. Suppose that you are an economist working for the CFTC and an exchange

has proposed to introduce futures trading on individual stocks. Would you accept this proposal? Give reasons for your answer. 10.8. Suppose that it is five minutes to the close of trading for the day and you

are a trader on the exchange floor. Your client gives you a sell order for one thousand contracts. You are holding one hundred contracts long. How can you take advantage of this situation? 10.9. Summarize what happened in the Hunts silver case. 10.10. Summarize what happened in the Salomon Brothers manipulation. 10.11. Why is the possibility of manipulation bad for the trading of futures contracts?

Give an example of what can go wrong. 10.12. Is selling an insurance contract speculating? Is buying an insurance contract

hedging? Are insurance contracts a benefit to society? Explain why or why not.

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10.13. Explain why speculation in futures trading is analogous to selling insurance

contracts. Does this imply that speculation with futures is beneficial or harmful to the economy? Explain your answer. 10.14. Do futures contracts need to be regulated? If yes, what can go wrong without

regulation? Give an example to explain your answer. 10.15. Can you identify common features in the manipulation stories involving

forward or futures markets? How can “authorities” intervene and break squeezes in a typical situation?

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10.16. What is banging the close?

11 The Cost-of-Carry Model 11.1 Introduction 11.2 A Cost-of-Carry Example 11.3 The Assumptions Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

11.4 The Cost-of-Carry Model The Model Setup Using the Law of One Price Using Nothing Comes from Nothing Different Methods for Computing Interest

The Arbitrage Table Approach

11.5 Valuing a Forward Contract at Intermediate Dates 11.6 Linking Forward Prices of Different Maturities 11.7 Summary 11.8 Cases 11.9 Questions and Problems

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11.1

Introduction

To trade a forward contract on a spot commodity and not get ripped off, one needs to know the fair forward price. But how can one determine this price? We claim that if you understand the concepts of present and future value introduced in chapter 2, then you should already be able to guess the answer. Give up? The answer is that the forward price should be the future value of the spot commodity today! Why? If all’s right with the world, then this should be today’s price for buying the spot commodity in the future. This chapter shows that under a reasonable set of assumptions, including the assumption of no arbitrage studied in chapter 6, this guess is correct. The technique we use to prove this guess is called the cost-of-carry model, and it is the simplest and first application of the arbitrage-free pricing methodology studied in chapter 6. Later chapters will use the same no-arbitrage approach to price more complex derivatives such as options. The cost-of-carry model is derived via a cash-and-carry argument whereby the underlying commodity is purchased with borrowed cash and held until the forward contract’s maturity date, thereby it is carried into the future. When held, various costs-of-carry (or carrying charges) such as interest on the borrowed cash and storage costs for the commodity are paid. At the forward contract’s maturity, the borrowing is repaid, yielding unfettered ownership of the commodity. The cost of purchasing the commodity in the future by the cash-and-carry strategy is easily determined. All of the ingredients are known at the start (the spot price, interest rates, storage costs). Because this strategy generates the same payoff as the forward contract, the forward price can be determined from the cost of constructing the strategy today; otherwise, an arbitrage opportunity exists. All’s right with the world after all! The reasonable assumptions underlying the cost-of-carry model are frictionless markets, no cash flows to the underlying commodity, no counterparty credit risk, and competitive markets. Interest rates and commodity prices are random. Financial assets that satisfy the no cash flow assumption include stocks that pay no dividends over the forward’s life and zero-coupon Treasury bonds that mature after the delivery date. The final results of this chapter use the cash-and-carry argument to value a forward contract at an intermediate date and to establish a link between two different maturity forward prices on the same commodity. We adopt a data-model-arbitrage approach that is used throughout the book. This approach starts with a numerical example (data). Next, we replace numbers with symbols (model). Finally, we generate arbitrage profits when securities are mispriced (arbitrage).

11.2

A Cost-of-Carry Example

The cost-of-carry model uses the concept of no arbitrage. We develop the basic cost-of-carry model via either of two equivalent techniques, which we labeled the law of one price and nothing comes from nothing in chapter 6. These simple techniques rest on several key assumptions. We discuss the assumptions after illustrating the ideas involved.

A COST-OF-CARRY EXAMPLE

Example 11.1 uses the law of one price. It considers two different ways of buying a stock in the future: buying a forward contract or buying the stock in the spot market and carrying it to the future. We create these two positions so that we can extract the forward price by equating their values today. We replace numbers with symbols to obtain the basic cost-of-carry model.

EXAMPLE 11.1: Finding the Forward Price Using the Law of One Price

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The Data ■

Today is January 1. Suppose that we go to the forward market and buy a one-year forward contract on Your Beloved Machine Inc. (YBM) stock. Unless noted otherwise, all computations are on a per stock basis. No cash changes hands today because the forward price F is the fair price. At maturity, we receive one YBM stock worth S(T) from the forward’s seller by paying him the forward price F dollars. Record the cash flows (in Table 11.1a) as 0 today (time 0) and S(T) – F on the delivery date (time T). This is Portfolio A, which we label as the market traded forward.

■

Alternatively, one can buy one share of YBM in the spot market today and carry it to the future. Assuming that YBM pays no dividends over the forward’s life, the cash flows (in Table 11.1b) are –100 today and S(T) on December 31.

■

To create a future liability equal to the forward price, borrow its present value by shorting zerocoupon bonds. This cash inflow reduces today’s net payment. If the interest rate is fixed at 6 percent per year, the cash flows are F/(1 + interest) = F/1.06 today and –F a year later (see Table 11.1b). These long stock and short bond positions make up portfolio B, which is our synthetic forward.

■

By the law of one price, to prevent arbitrage, portfolios A and B with identical future payoffs must have the same value today. Consequently, −100 + F/1.06 = 0 or F = 100 × 1.06 = $106

(11.1)

The forward price equals the amount repaid on the loan for the stock purchase.

TABLE 11.1A: Portfolio A: Long Market Traded Forward Portfolio

Today (Time 0) Cash Flow

Delivery Date (Time T) Cash Flow

Long forward (forward price F)

0

S(T) – F

Net cash flow

0

S(T) – F

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TABLE 11.1B: Portfolio B: Long Stock and Short Bonds (Long Synthetic Forward) Portfolio

Buy YBM in the spot market (stock price S = $100) Short zero-coupon bonds to borrow the present value of the forward price F (interest rate is 6 percent per year) Net cash flow

Now (January 1) Cash Flow

Maturity Date (December 31) Cash Flow

–100

S(T)

F/1.06

–F

–100 + F/1.06

S(T) – F

The Model ■

Use symbols to generalize. Replace 100 with S and 0.06 with R to get our first result (stated later as Result 11.1): F = S (1 + R) (11.2a) or S = BF

(11.2b)

where (1 + R) is the future value of $1 invested today (which is also known as the dollar return) and B ≡ 1/(1 + R) is today’s price of a zero-coupon bond that pays $1 at maturity.

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A Graphical Approach ■

■

■

One can also develop the cost-of-carry model using payoff diagrams. Recall from chapter 5 that these diagrams plot “Payoffs” (or gross payoffs) along the vertical axis and the “stock price on the delivery date,” S(T), along the horizontal axis. First, buy one YBM stock. Its payoff is a straight line starting at the origin and rising at a 45∘ angle. If the stock is worthless on the forward’s delivery date, then you get 0; if it’s worth $5, then you get $5, and so on (see Figure 11.1). Next, draw another diagram under this depicting the payoff from a short bond position with a future liability of F dollars. This is a line parallel to the horizontal axis that begins at –F on the vertical axis. Now, vertically add up the payoffs from these two graphs. The short bond trade lowers the long stock’s payoff by a parallel downward shift of F dollars. Consequently, the combined payoff starts at –F along the vertical axis, rises at a 45∘ angle to the horizontal axis at F, and keeps increasing. The combined position is a synthetic forward because its payoff on the maturity date is identical to that of a market-traded forward contract. Because the forward price F is chosen so that the traded forward contract’s value is zero, to prevent arbitrage, the synthetic forward must also have zero value. Setting the synthetic forward’s cost of construction (S – BF) to zero yields the cost-of-carry model.

Arbitrage Profits ■

What would you do if the forward price differs from the arbitrage-free price? Suppose that an errant trader quotes a forward price of $110. As this is higher than $106, sell the overpriced forward, buy

A COST-OF-CARRY EXAMPLE

the stock, and borrow the present value of $110. Then – 100 + 11/1.06 = $3.77 is your immediate arbitrage profit—and you have no future cash flow because the short traded forward and the long synthetic forward’s cash flows exactly offset each other. If the trader quotes $100 instead, reverse the trades and make –100/1.06 + 100 = $5.66. Any forward price other than the arbitrage-free price of $106 creates an arbitrage opportunity.

FIGURE 11.1: Cost-of-Carry Model Gross payoff

Long stock + Short bond

Long stock

0 S(T), stock price on delivery date Payoff 0

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S(T) Payoff similar to long forward

Short bond

−F Payoff

0 F

S(T)

−F

This example illustrates that there is nothing magical about the forward price. The forward price is linked to today’s spot price via the cost-of-carry model, which only includes the interest cost in this simple setting. This simple model raises several questions: 1. What are the hidden assumptions underlying this argument? The next section discusses the assumptions underlying this and most other models in this book.

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2. Is the cost-of-carry model useful in other contexts? In its simple form, it generates the forward prices for investment assets like currencies and stocks. Forward prices for consumption or production assets like corn, copper, and oil, which are primarily held for consumption or as inputs to production, may be obtained by modifying this model to incorporate additional carrying charges such as transportation and storage costs as well as convenience yield benefits. This extension is presented in Chapter 12. The remainder of this chapter generalizes this example to build the cost-of-carry model. First, we need to introduce the hidden assumptions underlying our arguments.

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11.3

The Assumptions

A model is a simplified or idealized description, often expressed mathematically, that helps us understand and explain reality. Financial models serve this purpose. Being abstractions, models rely on simplifying assumptions. To understand a model, one needs to understand the assumptions. First, they show when to use, when not to use, and how not to abuse a model. Second, they give a sense of when the model fails and the biases that may result. Understanding the assumptions changes a model from a black box to an open box. We need the following assumptions for derivative pricing models in general and for forward pricing models in particular. A1. No market frictions. We assume that there are no market frictions like transaction costs (brokerage commissions and bid/ask spreads), margin requirements, short sale restrictions, taxes that may be levied at different rates, and assets that are not perfectly divisible. The frictionless market assumption helps us create a benchmark model against which other models can be compared. Once you understand this ideal setting, you can add back frictions to better capture reality. In addition, it’s a reasonable first approximation for many institutional traders. For example, professional traders at major investment banks have far lower transaction costs than do small investors. Professionals often trade within the bid/ask spread, they have less stringent margin requirements, they can avoid short sale restrictions, and their trading profits are taxed at a uniform rate. Furthermore, frictionless markets may now be a reasonable first approximation for small traders in some markets because of the accumulated advances in information technology that have drastically reduced trading costs. A2. No credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty to a transaction will fail to perform on an obligation. We assume that there is no credit risk. If credit risk exists, cash flows become less predictable, and we stumble at the onset. No credit risk is a reasonable approximation for exchange-traded derivatives, whose performance is guaranteed by a clearinghouse. Vigilant exchange officials, alert brokers and dealers, the customer’s margin account, and the exchange’s capital minimize this risk. Over-the-counter derivatives, however, may possess substantial credit risks. In practice, prudent actions, such as trading only with counterparties

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THE ASSUMPTIONS

that have high credit ratings, having well-documented legal contracts, and requiring parties to keep collateral, reduce this risk. Pricing derivatives in the presence of credit risk is an area of active research. Jarrow and Turnbull (1995) developed a popular model for pricing credit derivatives that is used in industry, known as the reduced form approach. This extension is discussed in Chapter 26. A3. Competitive and well-functioning markets. In a competitive market, traders’ purchases or sales have no impact on the market price; consequently, traders act as price takers. In a well-functioning market, price bubbles are absent. We assume competitive and well-functioning markets. In a competitive market, a trader is individually too small for his trades to have a quantity impact on the price and to possess market power. Consequently, he acts as a price taker vis-à-vis the market. The competitive markets assumption is the workhorse of modern economics, and many powerful results are derived from it. As illustrated via the futures market manipulations described in chapter 10, this assumption does not always hold. Pricing securities when traders have market power and when prices can be manipulated is a difficult exercise. One must take into account bargaining and strategic interactions—ideas and concepts that are outside the scope of this book. Consequently, the cost-of-carry model is a poor model in such a setting. Focusing on competitive markets avoids these complications. We also assume that there are no asset price bubbles. A price bubble happens when an asset’s price deviates from its intrinsic or fundamental value. The fundamental or intrinsic value can be defined in two equivalent ways. The first is the present value of its future cash flows. The second is the price one would pay if, after purchase, you had to hold the asset forever. If you reflect on these, it becomes clear that they are equivalent definitions. A difference between the market price and the fundamental value only occurs if one believes that selling (retrading) can generate a higher value than holding forever—this difference is the bubble. The terminology reflects the popular belief that such deviations are short-lived and surely burst, like a soap bubble. It has been shown that many Internet stocks exhibited a price bubble during the 1990s, when they reached astonishingly high values despite an absence of dividend payments and continued reports of zero or negative earnings. We rule out price bubbles because our pricing arguments fail to hold in this context. Pricing derivatives in noncompetitive markets and in the presence of bubbles is a daunting task and a hot area of research. We return to this topic again in Chapter 19, when we discuss pricing options. A4. No intermediate cash flows. As a first pass, we assume that the underlying commodity has no cash flows over the forward contract’s life. This is a simplification that we will need to relax. When considering stocks, most companies reward their shareholders by paying dividends. US companies tend to be fairly conservative about their dividend policies. They pay fixed dividends and dislike tinkering with either the payment date or the amount, lest the market infers something negative from their actions. In addition, many commodities receive income or cash flows—coupon-paying bonds or consumption/production assets that provide convenience yields are prime

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examples. Moreover, consumption/production assets tend to incur transaction, storage, insurance, and other costs. For the present, we ignore these costs. A5. No arbitrage opportunities. This assumption’s meaning is self-evident. These five assumptions form the bedrock of our analysis, and they determine the validity of our models. Talented traders regularly monitor the market and check whether the assumptions are satisfied. If the assumptions aren’t satisfied, the pricing models and trading strategies may fail to work. In such a situation, one should not rely on the model’s implications. If one is using the model to hedge, the hedge may no longer work. An optimal strategy may be to close out the position, rethink, and come back another day. As in Kenny Rogers’s song, “you got to know when to hold ’em, know when to fold ’em, know when to walk away, know when to run.”

11.4

The Cost-of-Carry Model

We now assemble the different components that we have developed to formally derive our forward pricing model. If the notations seem abstract, remember the data from our example (S is $100, 1 + R = 1/B = $1.06) to get a numerical feeling for the argument.

The Model Setup

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The cost-of-carry forward pricing model has the following structure: ■

There are two dates. At time 0 (which is today), we start the clock, trade, and create one or more portfolios of securities. The clock stops at some later time T.

■

Three securities trade: a stock, a forward on the stock, and a zero-coupon bond. Unless specified otherwise, we use the terms asset, commodity, stock, and spot somewhat interchangeably.

■

The stock price is S(0) ≡ S today and S(T) at time T. The forward contract starts today and a forward price F(0, T) ≡ F is determined in the market such that no cash changes hands. This is the fair price. To understand why this price is fair, suppose the contrary. If it were not fair, then one side of the contract would be entering into the contract with an immediate loss and with no additional compensation.

■

The contract ends on the delivery or maturity date by cash settlement or physical delivery. The long’s payoff is S(T) – F and the short’s payoff is –[S(T) – F], which are equal and opposite.

■

Traders can borrow or lend funds at the risk-free rate by shorting or buying zerocoupon bonds. Let B(0, T) ≡ B be today’s price of a zero-coupon bond that pays one dollar at time T. Let 1 + R be the dollar return obtained at time T from investing one dollar in the zero today.

We utilize the law of one price (from Chapter 6) to formally derive the cost-ofcarry model.

THE COST-OF-CARRY MODEL

Using the Law of One Price To find the arbitrage-free forward price F, let’s consider two different ways of buying the stock in the future without a cash flow today. First, buy a forward contract. This cashless trade obliges you to acquire the stock by paying the forward price on the delivery date (this is our Portfolio A). Second, buy the stock today and finance this purchase by borrowing. This costless transaction gives you the stock and a liability equal to today’s stock price with its accrued interest on the same maturity date (this is our Portfolio B). These transactions are summarized as follows: Portfolio A (buy forward): Zero-cost portfolio (today) → Get the stock, pay the forward price F (write this as S[T] – F ) (maturity date) Portfolio B (buy the stock, sell the zero-coupon bonds to finance the stock purchase): Zero-cost portfolio (today) → Get the stock, repay the loan (write this as S[T] – S[1 + R]) (maturity date) Note that these two final payoffs only differ by the constant [F – S(1 + R)] and that this constant is known today. These two portfolios with identical initial values (zero) must also have the same value on the delivery date (see Figure 11.1). If one of these values differs, it opens up an arbitrage opportunity that will be exploited by vigilant traders standing ready to pick up money lying on the street—they will trade until the two prices are equal and no arbitrage remains. This implies that the forward price must equal to today’s stock price repaid with accrued interest, that is, [F = S(1 + R)].

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Using Nothing Comes from Nothing Instead of using the law of one price, if we subtract portfolio B from portfolio A, we could use the nothing comes from nothing principle instead. This technique creates a new portfolio that has a zero value in the future and therefore must also have zero value today. We show this trading strategy in Table 11.2, which shows the trades and arbitrage opportunity. As depicted, this portfolio is costless to create and worth [S(1 + R) − F] on the delivery date. To prevent arbitrage, a portfolio that is worthless today with a constant value at delivery must also have zero value on the delivery date. This is the nothing comes from nothing principle. The trading strategy is called a reverse cash-and-carry because it involves short selling the asset and making good on this obligation in the future. Alternatively, we could have done a cash-and-carry where we purchase the spot and carry it to the future by paying the necessary carrying costs. We skip the details because you can readily obtain the result by reversing the trades and changing the signs in Table 11.2. Figure 11.2 graphically shows the reverse cash-and-carry argument.

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TABLE 11.2: Arbitrage Table for Reverse Cash-and-Carry Portfolio

Today (Time 0) Cash Flow

Maturity Date (Time T) Cash Flow

Buy forward

0

S(T) – F

Short sell stock

S

–S(T)

Invest proceeds from short selling by buying risk-free zero-coupon bonds

–S

S(1 + R)

Net cash flow

0

S(1 + R) – F

FIGURE 11.2: No-Arbitrage Chain for Cost-of-Carry Model

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Now (Cash flow)

Maturity (Cash flow)

Buy forward Cost 0

Payoff S(T) – F How to get rid of S(T)?

Short sell stock Proceeds (+S)

Liability – S(T) Asset disappears

Lend proceeds Buy zero-coupon bonds worth S

Payoff S(1 + R)

Zero value (for sure)

The constant net payoff [S(1 + R) – F] must be zero (otherwise an arbitrage opportunity)

Different Methods for Computing Interest Remember from chapter 2 that we can compute B and R by using various discrete or continuous compounding methods. So far, we have computed B by using simple interest, which is the easiest case. We can also use continuous compounding by writing the zero-coupon bond price B or dollar return 1 + R as B = e−rT and 1 + R = 1/B = erT where r is the continuously compounded interest rate.

THE COST-OF-CARRY MODEL

We gather all our results together.

RESULT 11.1 Cost-of-Carry Model Consider a forward contract on a commodity beginning at time 0 (today) and maturing at time T. Then F = S (1 + R) (11.2a) Equivalently, S = BF

(11.2b)

where F ≡ F(0, T) is today’s forward price for delivery at time T, S ≡ S(0) is today’s spot price of the underlying commodity, 1 + R is the dollar return at time T from investing $1 in a zero-coupon bond today, and B ≡ 1/(1 + R) is today’s price for a zero-coupon bond that pays $1 at time T. Dollar returns are computed by the formulas Simple interest ∶ 1 + R = (1 + i × T) Continuously compounded interest ∶ 1 + R = erT

(11.2c) (11.2d)

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where i is the simple interest rate per year and r is the continuously compounded interest rate per year.

Continuously compounded interest rates give a better approximation than do simple interest rates when compounding is computed daily. Moreover, we will show in the next chapter that this assumption lays the foundation for developing a variety of cost-of-carry models, including those written on indexes and commodities that incur storage costs and have convenience yields. Example 11.2 uses this formula to compute forward prices and capture arbitrage profits.

EXAMPLE 11.2: Finding Forward Prices and Capturing Arbitrage Profits

The Forward Price ■

Consider the data from Example 11.1. The YBM stock price S is $100 today, a newly written forward on YBM matures in one year, and the simple interest rate is 6 percent per year. Then Equation 11.2c gives Dollar return 1 + R = (1 + 0.06 × 1) = $1.06 and the zero-coupon bond price B = 1/1.06 = $0.9434

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By expression (11.2a), the forward price is F = S (1 + R) = $106.00 which is the same price as in Example 11.1.1

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Capturing Arbitrage Profits ■

Suppose that an errant trader quotes a forward price of $110. Then the spot and forward prices are too far apart. The spot is $100, while the discounted forward price BF = $103.77 so that S – BF = – $3.77.

■

One can profit by doing a reverse cash-and-carry arbitrage. Simultaneously buy the relatively underpriced stock, sell the relatively overpriced forward, and short zero-coupon bonds with a maturity value of $110 to balance the portfolio. Immediately collect $3.77. The net payoff will be zero for sure on the maturity date.

■

Want to take your profits at maturity? Modify the bond position—short only enough zero-coupon bonds to finance the spot purchase, and make F – S(1 + R) = $4 on December 31.

■

Now, consider a different scenario—use the same data as before, except let the forward price be $100. Here S – BF = $5.66, suggesting that either the stock price is too high or the forward price is too low (or both). Do a reverse cash-and-carry arbitrage by selling the stock, buying the forward, and buying zero-coupon bonds. As before, the bond position can be tinkered with to take the profit now or later. If you buy zero-coupon bonds with a face value of $100, then you will make –BF + S = $5.66 of arbitrage profits today.

1 If this is continuously compounded interest, r is 6 percent, and then by (11.2a) and (11.2d), the forward price is F = S(1 + R) = 100 exp(0.06 × 1) = $106.18.

The Arbitrage Table Approach The previous examples motivate an arbitrage table approach for solving problems and proving results. By systematically recording cash flows at different dates, an arbitrage table enables direct comparison, which makes proofs easier. Although there are no set rules, we suggest three possible methods: 1. Understand the problem, using numbers if necessary. What do the different variables mean? Which security generates what cash flows? When? Draw a timeline. Plug dates and variable values into the formula to get a feel for the result to prove or how to make arbitrage profits. Use your intuition to sense which variables are overvalued or undervalued, using our “buy low, sell high” dictum. 2. Interpret cash flows. If the cash flows are positive today and zero on the delivery date, then you have made arbitrage profits. Alternatively, if cash flows are zero today and have a positive value on the delivery date, then you also have made arbitrage profits.

VALUING A FORWARD CONTRACT AT INTERMEDIATE DATES

3. Gather variables to one side of the equality and set the net cash flows at one of the dates to zero. It is inconvenient to work with nonzero cash flows at both the starting and ending dates. Life is easier if you gather variables to one side of an equality so that you can set today’s or the delivery date’s net cash flows to zero. Then, to prevent arbitrage, the other net cash flow must also be zero.

11.5

Valuing a Forward Contract at Intermediate Dates

Suppose you bought a forward contract some time back, but it still has remaining life. How much it is worth today? If the spot price soars or plunges, how would it affect the forward’s value? We use a cash-and-carry argument to develop a formula that answers these questions. The model may be used for mark-to-model accounting, which requires the valuation of assets and liabilities on a daily basis, a practice that is increasingly becoming popular with companies and regulators. Example 11.3 introduces a numerical example that is then generalized to obtain the relevant result.

EXAMPLE 11.3: Valuing a Forward Contract at an Intermediate Date

The Data

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■

Suppose that you bought a forward on January 1. Three months have passed. Meanwhile, the underlying price has increased. Of course, your long position has gained value because you entered into a contract to buy the underlying at a fixed price, but the underlying has become more valuable. What is the exact value of your long position today? Let us consider an example that uses some of our previous data: - On January 1 (time 0), YBM’s stock price S(0) was $100. - The simple interest rate i was 6 percent per year. - You purchased a newly written forward contract on YBM that matures in a year on December 31 (time T). - The forward price was set at F(0) = S(1 + R) = $106.

■

Three months have passed; today is April 1 (time t). YBM has rallied and the new stock price S(t) ≡ S is $120. You are still obliged to buy YBM by paying F(0) in nine more months (T – t is 0.75 years). But what about the value of your forward, V (t) ≡ V ? If you sell your position, how much should you get paid?

■

Draw a timeline to keep track of the cash flows (see Figure 11.3). The stock price increase has made the long forward position more valuable, the exact amount of which is determined by the arbitrage table (see Table 11.3).

■

Begin the portfolio construction as if you are buying the forward position by paying V. This has a payoff of S(T) – F(0) = S(T) – 106 on December 31. Next, short sell the stock. This gives $120 today but creates a liability –S(T) on the delivery date. The stock disappears, but you still have to come up

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with $106. This can be arranged by lending the present value of $106 via buying zero-coupon bonds. Notice that on April 1, the price of a zero-coupon bond that pays $1 on December 31 is B ≡ B (t, T) = 1/ (1 + 0.06 × 0.75) = $0.9569 ■

The portfolio has a zero value on December 31. To prevent arbitrage, −V + 120 − 0.9569 × 106 = 0 or V = $18.561

(11.3)

The Model ■

Replace numbers with symbols in Equation 11.3 to get −V + S − BF (0) = 0 or V = S − BF (0)

1

Instead, if we assume a continuously compounded interest rate r of 6 percent per year fixed, then F(0) = S(1 + R) = 100 × exp(0.06 ×exp(0.06 × 1) = $106.18, B(t, T) = exp(– 0.06 × 0.75) = $0.9560, and V = 120 – 0.9560 × 106.18 = $18.49. There is a 7 cent difference in value of a forward contract because of the different ways of computing interest.

This is Result 11.2, which is formally stated as follows.

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RESULT 11.2 Valuing a Forward Contract at an Intermediate Date Consider a forward contract created at time 0 and that matures at time T. On an intermediate date t (today), V = S − BF (0)

(11.4a)

where F(0) is the forward price at time 0 for a contract maturing at time T, V ≡ V (t) is today’s value of a long forward position, S ≡ S(t) is today’s spot price of the underlying commodity, and B ≡ B(t, T) is today’s (time t) price of a zero-coupon bond that pays one dollar at time T. In the case of simple interest, B is 1/[1 + i(T – t)], where i is the interest rate per year. In the case of continuously compounded interest, B is e–r(T − t) where r is the interest rate per year.

This is a more general version of our first result. If we set V = 0, as happens on a forward contract’s starting date, then Result 11.1 becomes a special case of Result 11.2. Using Results 11.1 and 11.2 together, we can obtain another insight. First,

VALUING A FORWARD CONTRACT AT INTERMEDIATE DATES

TABLE 11.3: Valuing a Forward Contract That Began Earlier Portfolio

Today, April 1 (Time t) Cash Flow

Buy forward (worth V (t) or V to long), forward price F(0) = $106

–V

Short sell spot (S = 120) Lend present value of forward price BF(0) Net cash flow

Delivery Date, December 31 (Time T) Cash Flow

S(T) – 106

120

–S(T)

–(1/1.045)106

106

–V + 120 – 0.9569 × 106

0

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FIGURE 11.3: Timeline for a Forward Contract That Began Earlier

Time 0

Forward started on January 1.

Time t (today)

Time T

B = 0.9569

$1

Today (April 1).

Forward matures one year from starting date.

Forward price Value of forward contract to long is V(t). Value of forward to long is S(T) – F(0). F(0) negotiated. Value of forward to long is 0.

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we note that Result 11.1, when applied to the time t forward price, implies that S = BF(t). Second, substitution of this fact into expression (11.4a) and some simple algebra yields an alternative expression for the value of the forward contract at time t: V = B [F (t) − F (0)]

(11.4b)

This expression shows that the value of the forward contract increases by the present value of the change in the new delivery price, F(t), relative to the original delivery price, F(0). The present value is needed to reflect the fact that the payments are not made until the delivery date, time T.

11.6

Linking Forward Prices of Different Maturities

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Consider a market in which the cost-of-carry model determines forward prices with high accuracy. If we change the delivery date, the model gives us a series of forward prices corresponding to contracts maturing on those different dates. We can also use the cash-and-carry argument to develop a formula linking these prices. This enables us to find one forward price when only the carrying cost and another forward price are known. When several forward contracts exist on the same commodity, this model also enables us to determine whether all the forward prices are correct or some of them are out of line. The model is given as our Result 11.3. A simple proof follows the result.

RESULT 11.3 Link between Near- and Distant-Maturity Forward Prices Consider two forward contracts that begin at time 0 (today), mature on the near-maturity date (time n) and the distant-maturity date (time d), respectively. Then, standing at time n, F (0, n) =

B (0, d) × F (0, d) B (0, n)

(11.5)

where F(0, n) and F(0, d) are the forward prices negotiated at time 0 for forward contracts maturing at times n and d, respectively, and B(0, t) is the price of a zero-coupon bond at time 0 that pays $1 at times t = n and t = d. The zero-coupon bond price B is 1/(1 + it) for simple interest and e–rt for continuously compounded interest, where i and r are the respective annual interest rates.

SUMMARY

Although we view this result as an application, it’s also a generalization. If you set the near-maturity forward to be at its delivery date n, then by convergence of the basis, F(0, n) becomes the spot price S(n), B(n, n) = 1, and expression (11.5) gives back Result 11.1 at time n. To prove Result 11.3, we just use Result 11.1 twice, once for time n and once for time d, getting the two equations S (0) = B (0, d) × F (0, d) S (0) = B (0, n) × F (0, n) Setting these two equations equal and solving for F(0, n) gives the result.

11.7

Summary

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1. We use no arbitrage via the law of one price and the principle that nothing comes from nothing introduced in chapter 6 to develop a forward pricing model. First, we create a securities portfolio to replicate a market-traded forward’s payoff. To avoid arbitrage, the cost of the replicating portfolio must equal to the forward contract’s arbitrage-free value. Equivalently, an alternative portfolio going long the forward contract and selling short the replicating portfolio (or vice versa) will have a zero value both now and in the future. Both approaches give the same price. 2. A portfolio consisting of a long stock, a short forward, and short some zero-coupon bonds can be created so that it has zero value on the forward’s maturity date. To prevent arbitrage, it must also have a zero value on the starting date. The forward price is determined from this condition. This gives Result 11.1, which is the basic cost-of-carry model. It states that the forward price is the future value of the spot price or F = S(1 + R). 3. The cost-of-carry model makes a number of assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no dividends over the contract’s life, and no arbitrage opportunities. 4. There are several ways of developing cost-of-carry models and solving arbitrage problems. We primarily use arbitrage tables, which systematically record cash flows at different dates. 5. We use the cash-and-carry argument to build more complex models. Result 11.2 states that the value of a forward contract at an intermediate date equals the spot price minus the discounted value of the old forward price or V = S – BF(0). Result 11.3 states that when two different forward contracts are written on the same commodity but with different maturity dates, the near-maturity forward price is a fraction of the distant-maturity forward price or F(0, n) = [B(0, d)/ B(0, n)] × F(0, d).

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11.8

Cases

American Barrick Resources Corp.: Managing Gold Price Risk (Harvard Busi-

ness School Case 293128-PDF-ENG). The case discusses derivative usage for hedging by a gold mining company. Banque Paribas: Paribas Derives Garantis (Harvard Business School Case

295008-PDF-ENG). The case explores issues connected with a broker-dealer setting up a derivatives subsidiary to achieve a credit rating. Hedging Currency Risks at AIFS (Harvard Business School Case 205026-PDF-

ENG). In this case a company considers managing foreign currency risks with derivatives.

11.9

Questions and Problems

11.1. Today’s price of gold in the spot market is $1,500 per ounce. The price of a

zero-coupon bond maturing in one year is $0.95. What will be the one-year forward price for gold? 11.2. The current price of Your Beloved Machine’s stock is $109. The continuously

compounded interest rate is 5.25 percent per year. What will be the fivemonth forward price for YBM stock? 11.3. The spot price of silver is $30 per ounce. The simple interest rate is 6 percent Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

per year. The quoted six-month forward price for silver is $31. a. What should be the arbitrage-free forward price for silver for a forward

contract maturing in six months? b. Demonstrate how you can make arbitrage profits in this market. 11.4. Today’s spot price of silver is $30 per ounce. The simple interest rate is

6 percent per year. The quoted six-month forward price for silver is $32. Transaction costs are $0.10 per ounce whenever spot silver is traded and a $0.25 per ounce one-time fee for trading forward contracts but no charges for trading bonds. a. Demonstrate how you can make arbitrage profits in this market if there are

no transaction costs. b. If you have to pay transaction costs, demonstrate how you will make

arbitrage profits or explain why you cannot make any such profits. 11.5. Suppose that the continuously compounded interest rate is 6 percent per year,

and the nine-month forward price for platinum is $1,750. What is today’s spot price of platinum? 11.6. What does the assumption of no market frictions mean? Is this assumption

true in current commodity markets?

QUESTIONS AND PROBLEMS

11.7. If the assumption of no market frictions is not true, then why should we study

models using this assumption? 11.8. What is the competitive market assumption, and what adverse market

behavior does it exclude? 11.9. What are the five assumptions underlying the cost-of-carry model for pricing

forward contracts? Which of these assumptions are most likely to be satisfied in current commodity markets? 11.10. State the key result of the cost-of-carry model in your own words. 11.11. Today is January 1. Forward prices for gold forward maturing on April 1 is

$1,500 per ounce. The simple interest rate is 6 percent per year. What would be the forward price for a forward contract on gold maturing on August 1? 11.12. Today is January 1. Forward prices for contracts maturing on April 1 and

on October 1 are $103 and $109, respectively. The simple interest rate is 8 percent per year. Assuming the spot price is $100 today, demonstrate two ways in which you can make arbitrage profits from these prices. 11.13. Today is January 1. Forward prices for contracts maturing on April 1 and

on October 1 are $103 and $109, respectively. On April 1, the price of a zero-coupon bond maturing on October 1 is $0.97. Assuming that the underlying interest rate is a constant interest rate, demonstrate one way of making arbitrage profits from these prices. 11.14. State the equation for the valuation of a forward contract in your own words.

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11.15. Suppose you bought a forward on January 1 that matures a year later. The

forward price was $214 at that time, and the simple interest rate was 7 percent per year. Six months have passed, and the spot price is now $150. What is the value of your forward contract today? 11.16. Suppose you trade a forward contract today that matures after one year. The

forward price is $105, and the simple interest rate is 7 percent per year. If, after six months from today, the spot price is going to be $150 and the value of the forward contract is $20, demonstrate how you can make arbitrage profit from these prices. 11.17. The value of a forward contract that you have been holding for the last six

months is $50 today. It matures in 3 more months. If today’s spot price is $100 and the underlying interest rate is 5 percent, what was the forward price that you had negotiated when you purchased the contract six months back? 11.18. Explain the difference between the forward price and the value of a forward

contract. How are they related? 11.19. What is the relation between forward prices of different maturities on the

same underlying? 11.20. (Excel) For gold and silver, collect from the internet today’s spot prices as well

as futures prices for futures contracts maturing up to one year. Also collect Treasury bill prices of different maturities. Using the above data, perform the

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following for three of the most actively traded futures contracts for gold as well as for silver: a. Assuming that interest is the only cost of carry, use Result 11.1 (Cost-of-

Carry Model) of Chapter 11 to compute forward prices that correspond to the maturities of three of the most actively traded contracts for the two metals. b. Compare these prices to the traded futures prices. Do they seem signifi-

cantly different? If so, explain why.

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Hints and Suggestions: ■

New York spot price for gold and silver can be found on a metal dealer’s (such as Kitco’s) website.

■

Futures prices can be obtained from an exchange’s (such as the CME Group’s) website www.cmegroup.com/market-data/index.html

■

Daily Treasury bill rates data can be found from the US Treasury’s website www.treasury.gov/resource-center/data-chart-center/interestrates/Pages/TextView.aspx?data=billrates

■

Use settlement prices wherever possible. If ask and bid prices are quoted for the spot or futures, take the average of the two.

■

Exclude futures contracts maturing in the current month as they can have liquidity problems.

12 The Extended Cost-of-Carry Model Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

12.1 Introduction 12.2 A Family of Forward Pricing Models 12.3 Forwards on Dividend-Paying Stocks The Cost-of-Carry Model with Dollar Dividends

12.4 Extended Cost-of-Carry Models 12.5 Backwardation, Contango, Normal Backwardation, and Normal Contango 12.6 Market Imperfections 12.7 Summary

The Cost-of-Carry Model with a Dividend Yield

12.8 Cases

A Synthetic Index

12.9 Questions and Problems

The Foreign Currency Forward Price

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12.1

Introduction

Cornell University is situated in rural upstate New York. The university campus is built on a hill overlooking beautiful Lake Cayuga and the city of Ithaca. The landscape is scenic, dotted with hills, valleys, and even a few gorges formed during the ice age. The winters are cold, snowy, and icy, which makes driving the roads treacherous. When choosing a car to own in Ithaca, a resident has many options to consider. Key among these options is the choice of rear-wheel, front-wheel, or four-wheel drive. To safely navigate the wintry roads, four-wheel drive is the preferred selection. Consequently, four-wheel drive cars are in great demand. Analogous to an Ithacan deciding which car to buy in hilly Ithaca, when constructing a forward pricing model, one needs to consider the market setting carefully. Commodities differ in whether they have cash flows, storage costs, or convenience yields. Choosing the correct model for proper pricing and hedging requires matching the model’s assumptions with market conditions. To do so otherwise leads to “treacherous driving” when using the model in practice. Chapter 11 presented the arbitrage-free forward pricing model under the assumption that the underlying commodity has no cash flows, storage costs, or convenience yields over the forward contract’s life. However, stocks and stock indexes pay dividends, bonds make coupon payments, foreign exchange prices are affected by both domestic and foreign interest rates, and physical commodities like corn or gold incur storage costs but may receive a convenience yield. The model in Chapter 11 also assumed frictionless markets and equal borrowing and lending rates, but trading involves brokerage commissions as well as the bid/ask spread, and borrowing and lending rates differ. How does one modify the forward pricing model when these assumptions are relaxed to get a better approximation? The purpose of this chapter is to answer this question. We do this by relaxing assumptions A1 and A4 of Chapter 11 and thereby incorporate dividends, storage costs, convenience yields, and market imperfections into our framework. Although this additional complexity appears daunting, it isn’t. The basic logic of the cost-of-carry model still applies, but with bells and whistles attached. Most of these extensions can be easily handled with only minor modifications to the basic model. By the end of this chapter, one should be able to choose the safest “vehicle” for forward pricing in realistic market “landscapes.”

12.2

A Family of Forward Pricing Models

The last chapter’s cost-of-carry model includes interest as the only cost-of-carry and invokes a cash-and-carry argument to express the forward price as the future value of the spot price (Result 11.1). As in Chapter 11, unless specified otherwise, we use the terms asset, commodity, stock, and spot interchangeably. Spot trading and a cash market transaction also have the same meaning. The cost-of-carry model relied on several critical assumptions that we repeat here for convenience: A1. No market frictions A2. No credit risk

A FAMILY OF FORWARD PRICING MODELS

A3. Competitive and well-functioning markets A4. No intermediate cash flows A5. No arbitrage opportunities This chapter studies the determination of forward prices in a variety of settings. Figure 12.1 provides a road map for the family of forward pricing models considered. We first generalize the model by relaxing the fourth assumption with the introduction of dividends. This gives the first three extended models of this chapter: ■

The simplest dividend model assumes that the underlying stock pays fixed dollar dividends on known dates (Result 12.1a). This model is useful for finding forward prices for contracts written on equities and coupon-bearing Treasury bonds that mature after the delivery date.

■

The second dividend model assumes that the underlying stock pays continuous dividends (Result 12.1b). It’s helpful for finding forward prices when the underlying is a stock index.

■

An important application of the second model is to determine forward prices for foreign currency forward contracts (Result 12.2).

FIGURE 12.1: A Family of Forward Pricing Models

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Cost-of-carry model (the basic model)

Link between forward prices of different maturities (an application)

Valuing a forward at an intermediate date With dividends

Currency forward (an application)

With storage costs and convenience yields

Generalized cost-of-carry

Backwardation Forward Price < Spot Price

Contango Forward Price > Spot Price

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The dividend model paves the way for richer models. One such extension is storage costs for holding a commodity mainly used for consumption or production like copper, corn, or crude oil. Storing the physical commodity may also provide a convenience yield like the ability to keep a production process running in times of temporary shortages. We include these costs and benefits to build further extensions of the model (Results 12.3 and 12.4). Last, we introduce market frictions such as brokerage costs, differential borrowing and lending rates, and limited access to proceeds from short sales. Here buys and sells are no longer symmetric. Well-defined prices give way to ranges within which arbitrage-free forward prices must lie (Result 12.5).

12.3

Forwards on Dividend-Paying Stocks

Most mature US companies pay their shareholders regular dividends (see Chapter 3 for a discussion of dividends). Companies dislike tinkering with the payment date or the dividend amount to minimize the possibility that markets will ascribe negative reasons for these actions and penalize the stock price. Firms like to wear the face of an English monarch from which you can read nothing. Consequently, fixed payment dates and fixed dollar dividends are reasonable assumptions for mature companies. Many assets, such as Treasury notes and bonds, have highly predictable cash flows and may be similarly modeled.

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The Cost-of-Carry Model with Dollar Dividends Consider a company that pays a fixed dollar dividend on a future date decided in advance. A dividend payment actually involves several dates including the declaration date and the report date. For our purposes, we only need to focus on the ex-dividend date, which is the cut-off for buying the stock with the dividend. Recall our discussion from Chapter 3. ■

Buying the stock before it goes ex-dividend gives the share plus the dividend; purchasing after this deadline gives the share without the dividend.

■

On the ex-dividend date, the cum-dividend stock price equals the ex-dividend stock price plus the dividend (see Result 3.1).

Dividends create an extra cash flow from holding the stock. If one adjusts the stock price for this cash flow, the standard argument follows. Example 12.1 shows how to do this. The resulting pricing model is identical to our cost-of-carry model, except that the stock price net the present value of the dividend replaces the stock price in the basic model.

FORWARDS ON DIVIDEND-PAYING STOCKS

EXAMPLE 12.1: The Forward Price when the Underlying Pays a Fixed Dollar Dividend

The Data ■

Consider the data from Example 11.1. Your Beloved Machine Inc.’s (YBM) stock price S is $100 today (January 1, time 0), and the simple interest rate is 6 percent per year. Consider a newly written forward contract on YBM that matures in one year (December 31, time T).

■

Now, assume that the stock pays a dividend div of $1 in three months (April 1, time t1 ). Figure 12.2 gives the timeline for these cash flows. Table 12.1 accommodates this dividend by adding an extra column to our arbitrage table.

■

To determine the forward price, we use the nothing comes from nothing arbitrage principle. We create a portfolio that has zero net payoffs on all future dates. To handle the dividend, we need to borrow cash to generate a future liability just equal to the dividend payoff. Here are the details: - Buy one share of YBM. This gives a dividend div = $1 after three months and a stock worth S(T) after one year. - Zap the dangling dividend by borrowing its present value. As B1 = 1/(1 + 0.06 × 0.25) = $0.9852 is the price of a zero-coupon bond maturing in three months, record the cash flows as B1 div = $0.9852 today and –$1 on the ex-dividend date. - Sell YBM forward to eliminate it from the portfolio on the maturity date. - Finally, get rid of the forward price F by borrowing its present value. Jot down the cash flows as BF = (1/1.06)F today and –F after one year.

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■

To prevent arbitrage, today’s net cash flow must also be zero: −100 + 0.9852 × 1 + 0.9434F = 0 or, F = $104.96

(12.1)

The Model and Arbitrage Profits ■

Replace numbers with symbols in (12.1) and rearrange terms to get S − B1 div = BF

■

(12.2)

Suppose a dealer forgets to adjust for the dividend and quotes a forward price of $106. To obtain arbitrage profits, create the portfolio given in Table 12.1. This has a short position in the overpriced dealer-quoted forward and a long position in the synthetic forward. Then −S + B1 div + BF = −100 + 0.9852 + (106/1.06) = $0.9852 The present value of the dividend is the immediate arbitrage profit.

■

If the quoted forward price is less than the arbitrage-free price, reverse the trades to capture the arbitrage profit.

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Formalizing this example gives our first result.

RESULT 12.1A Cost-of-Carry Model with Dividends Consider a forward contract written today on a stock that pays a fixed dollar dividend div on a known future date. Then S − B1 div = BF

(12.2)

where today (time 0) ≤ ex-dividend date (time t1 ) ≤ maturity date (time T), S is today’s spot price, F is today’s forward price for time T, and B1 ≡ B(0, t1 ) and B ≡ B(0, T) are today’s prices of zero-coupon bonds maturing on the ex-dividend and the delivery dates, respectively. The zero-coupon bond price B is 1/(1 + iT) in the case of simple interest and e−rT for continuously compounded interest, where i and r are the annual interest rates. B1 is computed by replacing T with t1 in these expressions.

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FIGURE 12.2: Timeline of Cash Flows for a Dividend Paying Stock

January 1 Time 0 (Today) B1 = 1/1.015 B = 1/1.06 Forward starts.

December 31 Time T (One year later)

April 1 Time t1 (Three months later) 1 1 YBM pays dividend div of $1 to the stockholder.

Forward matures. Long’s payoff is S(T) – F.

FORWARDS ON DIVIDEND-PAYING STOCKS

TABLE 12.1: Arbitrage Table for Finding the Forward Price of a DividendPaying Stock Portfolio

Buy stock, get dividend div after three months Borrow present value of the dividend B1 div today Sell forward Borrow present value of forward price BF Net cash flow

Today, January 1 (Time 0) Cash Flow

–100

(1/1.015) × 1

Dividend Date, April 1 (Time t1 ) Cash Flow

Delivery Date, December 31 (Time T) Cash Flow

1

S(T)

–1

0

–[S(T) – F]

(1/1.06)F

–100 + 0.9852 × 1 + 0.9434F

–F

0

0

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We can easily extend this to accommodate more dividends. Suppose that the stock pays another dividend div2 at time t2 before the forward matures. Then, the result gets modified to S − (B1 div1 + B2 div2 ) = BF (12.3) where div1 has a subscript to link it with the payment date and B2 is today’s price of a zero maturing on the second ex-dividend date. In its most general form, S − (Present value of all dividends over the forward’s life) = BF

(12.4)

Why adjust for dividends? We do these adjustments because dividends differently impact the stock and the forward: the long stock earns dividends but the long forward doesn’t. Once you take out the dividends accruing to the stock, it puts both positions on the same footing, and our familiar cost-of-carry argument applies. Next, we introduce forwards on indexes. For these contracts, the dividend adjustment is more complex.

The Cost-of-Carry Model with a Dividend Yield Modern portfolio theory (MPT; say it quickly and it comes out as “empty!”) pioneered the idea of investing in diversified portfolios that replicate the returns on stock market indexes. An index mutual fund (or a stock index fund) pursues this objective by holding a basket of securities in the same proportion as the index. The fund’s share price is usually determined at day’s end. An exchange traded fund (ETF) strives to do the same, but it trades continuously on an exchange.1 Indexes and index funds underlie a large number of derivative securities, including forward contracts.

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What is the arbitrage-free forward price of such an indexed portfolio? Unfortunately, the dividend adjustment method outlined in the previous section, which assumes that the dividends and their payment dates are nonrandom, works poorly for an index because there are too many dividends. This section, therefore, develops an alternative approach that accommodates random dividends and random payment dates. This modification uses dividend yields and the continuously compounded version of the cost-of-carry model (see Chapters 3 and 11, respectively). To apply this modification, we first need to understand how to adjust an index’s value for continuously paid dividends. And this, in turn, requires a discussion of synthetic index construction.

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A Synthetic Index An index is a portfolio of stocks (or assets) formed according to some rules. A synthetic index is a portfolio created for the purpose of replicating the returns on a traded index. Indexes come in two varieties: those that adjust for dividends and those that do not. Total return indexes assume that all disbursements from the portfolio’s stocks, including regular dividends, get reinvested in the index portfolio. For total return indexes, the reinvestment assumption effectively removes the dividends from consideration because there are no cash flows (see Figure 12.3). In this case, forwards on total return indexes can be priced using the no cash flow assumption, and the results of Chapter 11 apply. However, most indexes, including the Dow Jones Company’s and Standard and Poor’s, make no adjustment for regular cash dividends. In this case, dividends need to be explicitly considered. The trick is to compute the value of the index today (time 0), excluding the present value of the dividends paid over the life of the forward contract. To compute this adjustment, it is helpful first to consider how such an indexe’s value changes if all dividends are reinvested, as in a total return index. Let the value of the index today without reinvestment be S(0) and its value at time T be denoted S(T). Assume that it pays a continuous dividend yield of 𝛿 percent per year. For comparison purposes, consider an otherwise identical index with the same time 0 value I(0) = S(0). For this index, however, assume that all of the continuously paid dividends are reinvested back into this index. Let us call this the synthetic total return index. Because the dividends are reinvested into I(T), its value at time T will exceed that for the index where dividends are not reinvested, that is, I(T) > S(T). In fact, using Result 3.2, we know that I (T) = S (T) eᄕT Note that S(0) represents the present value of the index at time T plus the dividends. To get the present value of the index, less the continuously paid dividends over the life of the forward contract, we just remove the dividend yield term; that is, present value of S (T) = S (0) e−ᄕT

FORWARDS ON DIVIDEND-PAYING STOCKS

This expression gives us the modified “stock price” to use in the forward pricing model. Notice that this is a quantity adjustment in contrast to the price adjustment used previously for dollar dividends. This modification assumes that the dividends on the index are paid over the year at a uniform rate (see Figure 12.3). Although some dividends are smaller, while others are larger, this is a reasonable assumption if there are sufficiently many stocks in the index and dividends are paid randomly over time. Newspapers and other information vendors regularly report these dividend yields. Example 12.2 uses this stock price modification to determine the forward price on an index. In practice, an index derivative usually has a multiplier, for example, the Chicago Mercantile Exchange’s S&P 500 futures is $250 times the S&P 500 index level, and if the previous day’s futures price is 1,000 and today’s price is 1,000.10, then the long would have 250(1,000.10 – 1,000) = $25 credited to her brokerage account. As we have done before, we ignore the multiplier and do calculations on a per unit basis.

EXAMPLE 12.2: The Forward Price for an Index with a Dividend Yield

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The Data ■

Consider a fictitious “INDY index” obtained by averaging stock prices and a synthetic index “INDY spot” that replicates its performance. INDY’s current level S is 1,000, and the index does not reinvest dividends. A newly written forward contract on the index matures in a year (time T). Let the continuously compounded interest rate r be 6 percent per year.

■

Stocks constituting INDY index paid $20 of dividends last year and are expected to pay the same this year. Then, the dividend yield 𝛿 = 20/1,000 = 0.02 or 2 percent per year.

■

■

If you buy e−𝛿T = e−0.02×1 = 0.9802 units of INDY spot and reinvest the dividends on a continuous basis, then you will have one unit of the index worth S(T) at year’s end. Table 12.2 records the following trades in an arbitrage table: - Sell the forward contract on the INDY index. This gives a zero value at time 0 and –[S(T) – F] on the delivery date T. - Buy 0.9802 units of INDY spot, and record the cash flows as – 0.9802 × 1,000 = –980.20 today. Reinvest all dividends until delivery back into the index to get S(T) on the maturity date. - Borrow the present value of F. A zero-coupon bond maturing one year later is worth B = e−rT = e−0.06 = $0.9418 today. Record the cash flows as BF = 0.9418F on January 1 and –F on December 31.

■

The resulting portfolio has a zero payoff on the delivery date. To prevent arbitrage, the net payoff is zero today. Consequently, −0.9802 × 1, 000 + 0.9418F = 0 or F = $1, 040.81

(12.5)

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The Model and Arbitrage Profits ■

Replace the numbers with symbols in Equation 12.5 to get −e−𝛿T S + e−rT F = 0 or F = Se(r − 𝛿)T

■

Suppose an errant dealer quotes $1,020 as the forward price for INDY index. As this is less than the arbitrage-free price, buy the relatively underpriced traded forward and sell short the synthetic forward as in Table 12.2. This gives an immediate arbitrage profit of −e−rT F + e−𝛿T S = −0.9418 × 1, 020 + 0.9802 × 1, 000 = $19.60

FIGURE 12.3: Creating a Synthetic Total Return Index Today (Time 0)

Maturity (Time T)

(a) Total return index

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I(0)

…

…

I(T)

All dividends reinvested in the index. (b) Most indexes S(0)

…

…

S(T)

Underlying stocks pay numerous dividends, but the index is not adjusted. Now (Time 0)

End Date (Time T)

(c) Synthetic total return index S(0)

…

…

S(T)eδT

Reinvest all dividends as paid S(T) e−δT Spot)

Forward Price

Backwardation (Forward price < Spot)

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FIGURE 12.6: Normal Contango and Normal Backwardation Normal Contango

Forward Price Speculators (Net)

5.05

Hedgers (Net)

5 Short Contracts

–500 0

500

Long Contracts

Expected spot price is $5 Forward price is $5.05 in equilibrium (net demand = net supply)

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Normal Backwardation Farmer

Forward Price

Hedgers (Net)

5 4.95

Short Contracts

–450 0 450

Expected spot price is $5 Forward price is $4.95 in equilibrium

Speculators (Net) Long Contracts

MARKET IMPERFECTIONS

Consider another scenario where corn farmers, who want to hedge by selling forward and fix a selling price for their corn, dominate the forward market (see Figure 12.6). These hedgers would like to go short. Their demands are met by speculators who are willing to go long only if the expected spot price exceeds the current forward price. This is because for the speculators to be in the market, their expected return must be positive. By selling at a price below the expected future spot price, hedgers entice the speculator to go long, and this transfers the unwanted risk to the speculator. This creates a normal backwardation because the forward price is less than the expected future spot price. And, the risk premium is reversed. The concepts of normal contango and normal backwardation are, thus, exactly the notions needed to understand risk premium and whether the forward price exceeds or is less than the expected futures spot price of the commodity. Our cost-of-carry models cannot answer these more subtle questions. The trade-off is that the cost-ofcarry model gives a price, but the economic approach doesn’t.

12.6

Market Imperfections

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How does the cost-of-carry model get modified if we start considering market “imperfections” such as transaction costs, different borrowing and lending rates, and restrictions on short selling? Here the cash-and-carry arbitrage and the reverse cashand-carry are no longer symmetric. Consequently, we get price ranges within which the forward price lies. These bounds are likely close because it’s the trader with the lowest transaction costs (and not retail traders like us) who take advantage of arbitrage opportunities and determine the price bounds. Example 12.5 introduces brokerage commissions into our basic cost-of-carry model.

EXAMPLE 12.5: Forward Price Bounds in the Presence of Market Imperfections ■

Consider the data from Example 11.1. YBM’s stock price S is $100 today (January 1, time 0), and the dollar return (1 + R) is $1.06. Consider a newly written forward on YBM that matures in one year (time T).

■

Assume that when trading YBM, a brokerage commission is charged that equals TC = 1 percent of the stock price. For simplicity, assume that there are no transaction costs on the maturity date. Now we need two tables, Tables 12.5a and 12.5b, for an arbitrage portfolio.

■

Table 12.5a shows, to prevent arbitrage, −101 + 0.9434F ≤ 0 or F ≤ 107.06 This gives an upper bound for the forward price.

(12.15a)

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■

Table 12.5b shows the no-arbitrage portfolio for a reverse cash-and-carry. Here −0.9434F + 99 ≤ 0 or 104.9396 ≤ F

(12.15b)

This gives a lower bound for the forward price. ■

Combining expressions (12.15a) and (12.15b), 104.9396 ≤ F ≤ 107.0596

(12.15c)

The forward price lies between $104.9396 and $107.06. ■

Replacing numbers with symbols gives the formula S (1 + R) (1 − TC) ≤ F ≤ S (1 + R) (1 + TC)

(12.16a)

TABLE 12.5A: Arbitrage Table for Cash-and-Carry Portfolio

Buy spot (pay 1 percent transaction cost today)

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Sell forward Borrow present value of F Net cash flow

Today (Time 0) Cash Flow

Maturity Date (Time 1) Cash Flow

–100 × (1 + 0.01)

S(T)

0

–[S(T) – F]

0.9434F

–F

–101 + 0.9434F

0

TABLE 12.5B: Arbitrage Table for Reverse Cash-and-Carry Portfolio

Sell spot (deduct transaction costs from short sale proceeds) Buy forward Lend present value of F Net cash flow

Today (Time 0) Cash Flow

Delivery Date (Time 1) Cash Flow

100 × (1 – 0.01)

–S(T)

0

S(T) – F

–0.9434F

F

–0.9434F + 99

0

MARKET IMPERFECTIONS

Besides brokerage commissions, there may be a bid/ask spread.4 As transaction costs increase, the price bounds become farther apart. However, different traders may have different transactions costs. This opens up quasi-arbitrage (near-arbitrage) opportunities for the lowest cost trader. One can introduce more market frictions in a similar fashion. For example, with unequal borrowing and lending rates, the relation becomes S (1 + RLEND ) ≤ F ≤ S (1 + RBORROW )

(12.16b)

where 1 + RLEND is the dollar return from lending and 1 + RBORROW is that for borrowing. Several of these relations may be combined.

RESULT 12.5 The Cost-of-Carry Model in the Presence of Brokerage Commissions and Unequal Borrowing and Lending Rates When stock trades are charged a brokerage commission of TC on today’s trade, and the borrowing rates are higher than the lending rates, then the forward price F satisfies:

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S (1 + RLEND ) (1 − TC) ≤ F ≤ S (1 + RBORROW ) (1 + TC)

(12.16a)

where S is today’s stock price, F is today’s forward price for time T, and 1 + RLEND and 1 + RBORROW are the dollar returns for lending and borrowing, respectively.

Like playing with children’s blocks, you can combine Results 12.4 and 12.5 to build even more general models. To apply the cost-of-carry model in practice requires a good understanding of the market and matching the market’s structure to the model’s assumptions. For financial commodities like stock indexes or Treasury securities, convenience yields and storage costs are not relevant, so the simpler models apply. In contrast, for agricultural commodities like corn or wheat, convenience yields and storage costs are extremely important and need to be included. Given the previous discussion, all that is left for a correct application of these models is experience and market savvy. Both of these will come in good time, and the old saying is especially relevant in this regard: “practice makes perfect.”

4

And when you sell short, the brokerage firms may not allow you the full proceeds—you may only earn interest on part of the funds kept with the broker. When there are restrictions on short selling, you may have an inequality like S(1 + q × Interest) ≤ F ≤ S(1 + R), where q is the fraction of usable funds derived from the short sale. Notice that this arbitrage only works in one direction because long and short positions have asymmetric payoffs. This relationship is less important because rules of short selling tend to vary from country to country.

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12.7

Summary

1. We extend the basic cost-of-carry model to incorporate market frictions (relax assumption A1 of Chapter 11) and dividends (ease A4) into the framework. 2. Assuming that interest is the only relevant carrying charge, we get three forward pricing models for assets that generate income:

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a. The simplest dividend model assumes that the underlying stock pays fixed dollar dividends on known dates. This gives Result 12.1A, which states that the stock price net of the present value of dividend is the discounted forward price or S – B1 div = BF. This is useful for finding forward prices for contracts written on dividend-paying stocks and coupon-bearing Treasury bonds that mature after the delivery date. b. When the underlying stock pays dividends at a continuous rate 𝛿, we get Result 12.1B: the forward price is the compounded value of the stock price, where compounding is done at a net rate equal to the difference between the interest rate and the dividend yield or F = Se(r − 𝛿)T . This is useful for finding forward prices when the underlying is an index portfolio. c. A modification of the previous model gives covered interest rate parity, which expresses the forward exchange rate as the spot exchange rate compounded at a net rate equal to the difference between the domestic and the foreign interest rates. Result 12.2 states that FA = SA (BE /B) = SA e(r − rE )T . 3. For commodities mainly used for consumption or production, storage costs can be important. However, holders of the physical commodity may also get a convenience yield. We include these costs and benefits to build a generalized costof-carry model (Result 12.4): the forward price is the compounded value of the spot price, where compounding is done at a net rate equal to the costs (interest rate r plus the storage cost rate g) minus the benefits (dividend yield 𝛿 plus the convenience yield y) or F = Se((r + g)−(𝛿 + 𝛾))T . 4. Contango is a market where the forward price is higher than the spot price. This happens when the net cost-of-carry is positive in the generalized cost-of-carry model. The market is in backwardation when the reverse happens. 5. Normal contango happens when a majority of the hedgers want to set up a long hedge. Speculators assume the other side of the trade, but they require a risk premium for supplying liquidity: the forward price is greater than the expected future spot price. When the net hedgers are short, the risk premium is reversed, and a normal backwardation market is created where the forward price is less than the expected future spot price. 6. Buys and sells are no longer symmetric when market imperfections like brokerage costs, differential borrowing and lending rates, and short selling restrictions are introduced. Well-defined prices give way to ranges within which forward prices lie.

QUESTIONS AND PROBLEMS

12.8

Cases

Diva Shoes Inc. (Darden School of Business Case UV0265-PDF-ENG, Harvard

Business Publishing). The case studies a US-based manufacturer’s currency risk exposure and considers whether hedging via a forward contract or a currency option is advisable. Leveraged Buyout (LBO) of BCE: Hedging Security Risk (Richard Ivey School

of Business Foundation Case 908N23-PDF-ENG, Harvard Business Publishing). The case considers an equity partnership’s currency risk exposure and evaluates various derivative instruments for hedging those risks. Risk Management at Apache (Harvard Business School Case 201113-PDF-ENG).

The case evaluates a company’s hedging strategy and the derivatives used for this purpose.

12.9

Questions and Problems

12.1. a. Explain the cost-of-carry model with dollar dividends in your own words. b. Justify why the spot price considered in the model is net of the present value

of all future dividends paid over the life of the contract. c. Why don’t we adjust for dividends that are paid after the forward’s maturity

date?

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12.2. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays a dollar

dividend after two months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.3. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.4. Boring Unreliable Gadget Inc.’s stock price S is $50 today. The company,

however, reliably pays quarterly dividends to shareholders. For example, BUG paid $0.95 dividend one month back; it will pay $1 dividend two months from today, $1.05 after five months, $1.10 after eight months, and $1.15 after eleven months. If the continuously compounded interest rate is 4 percent per year, what is the forward price of a six-month forward contract on BUG? 12.5. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. The continuously compounded interest rate is 4 percent per year. If the six-month forward price is $51, demonstrate how you can make arbitrage profits or explain why you cannot.

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12.6. Boring Unreliable Gadget Inc.’s stock price S is $50 today. It pays dividends

of $1 after two months and $1.05 after five months. The continuously compounded interest rate is 4 percent per year. Transactions costs are $0.10 per stock traded, a $0.25 one-time fee for trading forward contracts, and no charges for trading bonds. If the six-month forward price is $51, demonstrate how to make arbitrage profits or explain why you cannot. 12.7. Explain how the cost-of-carry model with dollar dividends differs from the

cost-of-carry model with dividend yields. 12.8. What is a convenience yield for a commodity and why is it important to

include in the cost-of-carry model? Give an example of a commodity that has a convenience yield. 12.9. a. What is a stock index? What happens to stock indexes when dividends are

paid? b. What is a synthetic index? Why is it useful? 12.10. Consider SINDY Index obtained by averaging stock prices and a synthetic

index SINDY spot that replicates its performance. (1) SINDY’s current level I is 10,000, and the synthetic index’s price S is $10,000. (2) A newly written forward contract on the index matures after T = 0.5 years. (3) The continuously compounded interest rate r is 5 percent per year. (4) Stocks constituting SINDY spot paid $190 of dividends last year and are expected to pay the same this year. a. Compute the dividend yield 𝛿 on SINDY spot.

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b. Compute the forward price F. 12.11. Consider the data given in problem 12.11 for futures contracts on SINDY

Index. If the forward price is F = $10,231, show how you can make arbitrage profits or explain why you cannot. 12.12. Consider the data given in problem 12.11 for futures contracts on SINDY

Index. Suppose you have to pay transaction costs: (1) When you go long or sell short a portfolio of stocks, transaction costs (brokerage fees plus the price impact of the trade) equal 10 basis points of the synthetic index’s price. (2) There is a one-time fee of $15 for trading forward contracts but no charges for trading bonds. If the forward price is F = $10,231, show how you can make arbitrage profits or explain why you cannot. 12.13. Today’s spot exchange rate SA is $1.30 per euro in American terms. The

continuously compounded annual risk-free interest rates are r = 4 percent in the United States (domestic) and rE = 3 percent in the Eurozone. What is the four-month forward rate in American terms if the cost-of-carry model holds?

QUESTIONS AND PROBLEMS

12.14. The current dollar/Swiss franc spot exchange rate is 0.5685. If you invest one

dollar for ninety days in the US domestic riskless asset, you earn $1.0101, and if you invest one franc for ninety days in the Swiss riskless asset, you earn 1.0113 francs (assume continuous compounding). A broker offers you a ninety-day forward contract to buy or sell 1 million francs at the exchange rate of 0.55 dollars/franc. Are there arbitrage profits to be made here? If so, compute them. 12.15. Alloyum costs $0.10 per month to store (which is paid upfront) but gives a

convenience yield of $0.12 per month (which is received on the maturity date). If its spot price S is $200 per ounce and the continuously compounded interest rate r is 5 percent per year, what is the five-month forward price for Alloyum? 12.16. In the previous example, suppose a trader quotes a five-month forward price

of $206 per ounce. Demonstrate how you can make arbitrage profits from these prices; if you cannot, explain your answer. 12.17. COMIND Index is computed by averaging commodity prices. Compute

the five-month forward price for this index if the spot price is 1,000 and the continuously compounded annual rates for various costs and benefits are 5 percent for the interest rate, 3 percent for the dividend yield, 4 percent for the storage cost, and 3 percent for the convenience yield. 12.18. Find the price bounds for the five-month forward price for Boring Unreliable

Gadgets when

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a. BUG’s stock price S is $50 today b. a trader can borrow money at 5 percent and lend money at 4 percent,

where the interest rates are annual simple interest rates c. a brokerage commission of 0.5 percent of the stock price is charged today

but your broker waives transaction costs on the maturity date 12.19. Forward prices for April and June forward contracts for platinum are $400 per

ounce and $410 per ounce, respectively. The interest rate is 1 percent for the April–June period. There is a 1 percent transaction cost whenever you trade (buy or short sell) the spot in June, but the broker is waiving any transaction costs for trading forward contracts and brokerage costs for trading the spot in April. Can you generate arbitrage profits? Explain your answer. (Assume that the spot price of platinum in June lies between $400 and $500 per ounce.)

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13 Futures Hedging 13.1 Introduction

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13.2 To Hedge or Not to Hedge

13.5 Futures versus Forward Hedging

Should the Firm or the Individual Hedge?

13.6 Spreadsheet Applications: Computing h

The Costs and Benefits of Corporate Hedging

13.7 Summary

EXTENSION 13.1 Airlines and Fuel Price Risk EXTENSION 13.2 A Hedged Firm Capturing a Tax Loss

13.3 Hedging with Futures

13.8 Appendix Deriving the Minimum-Variance Hedge Ratio (h) Computing the Minimum-Variance Hedge Ratio (h)

Perfect and Imperfect Hedges

Statistical Approach

Basis Risk

Econometric Approach: A Linear Regression Model

Guidelines for Futures Hedging

13.4 Risk-Minimization Hedging The Mean-Variance Approach Limitations of Risk-Minimization Hedging

13.9 Cases 13.10 Questions and Problems

TO HEDGE OR NOT TO HEDGE

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13.1

Introduction

The novelist Fyodor Dostoyevsky perceptively observed in The Gambler: “as for profits and winnings, people, not only at roulette, but everywhere, do nothing but try to gain or squeeze something out of one another.” This aptly defines a zero-sum game of which, excluding transaction costs, futures trading is an excellent example. There is a pool of talented futures traders who earn a living from this business thereby depressing the chances of winning for ordinary traders who lose money on average. Then, why trade futures? Cynics give a facile explanation: people are “into commodities” because it’s as thrilling as going to the casinos. True to circus-owner P. T. Barnum’s immortal quote that “there’s a sucker born every minute,” these gamblers gladly lose for the joy of the ride. Perhaps others believe that they really can win the futures game. We know no surefire strategy. Even if you stumble on such a strategy, repeated use will destroy its efficacy. So, why trade futures? In reality, most futures contracts are traded to hedge risks that preexist in some line of business, and hedgers are willing to give up expected returns as payment for this “insurance.” We discussed this issue in the context of normal backwardation in Chapter 12. For example, Figure 13.1 shows a man selling goods on some tourist spot, perhaps a beach. If he chooses his wares wisely, say, by carrying an assortment of umbrellas and sunglasses, then he has hedged well. Come rain or come shine, he has something to sell. If this is not possible, he could use futures to manage these risks. For example, he could sell only umbrellas and buy futures on a sunglass company’s stock, instead of selling sunglasses. Who hedges? A majority of the Fortune 500 companies and a growing number of smaller firms hedge risk. Farmers often hedge the selling price of their produce. Many producers hedge input as well as output price risk. This chapter explores the reasons for hedging, discusses its cost and benefits, and introduces futures hedging strategies. We discuss perfect and cross hedges, long and short hedges, and risk-minimization hedging using standard statistical techniques.

13.2

To Hedge or Not to Hedge

Unfortunately, there is no general answer to what may be phrased as (with apologies to the immortal William Shakespeare) to hedge or not to hedge—that is the question. We can only discuss hedging’s benefits and costs.

Should the Firm or the Individual Hedge? The classic Modigliani and Miller (M&M) papers argued that debt policies do not affect firm value. This irrelevance follows because a shareholder can replicate such policies herself by trading stocks in otherwise identical firms. Therefore the noarbitrage principle ensures that these companies, differing only in their debt policies, must have equal value. The same argument can be applied to a firm hedging its balance sheet risks giving intellectual support to the irrelevance of hedging. However, the M&M results rely on several key assumptions, including no market frictions

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FIGURE 13.1: Wisely Hedged: Come Rain, Come Shine, He Has Something to Sell*

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Umbrellas $10

Umbrellas $10 Sunglasses $15 Inspired by a 1980s newspaper advertisement.

(including bankruptcy costs), no taxes, and no information asymmetry. In the real world, these assumptions fail to hold. Consequently, if reducing the firm’s risk is in the shareholders’ best interest, then a firm can usually do a better job of hedging than individual investors. Indeed, a company (1) is likely to have lower transaction costs; (2) can trade larger contracts than a shareholder; (3) can dedicate competent personnel to hedging; (4) can hedge by issuing a whole range of securities, which individuals cannot; (5) may have private information about the company’s risks; and (6) can hedge for strategic reasons that lie beyond an ordinary investor’s knowledge. In all these cases, a company creates more value than an individual investor hedging on her own. But hedging is costly to implement, both in developing and retaining the relevant expertise within the firm, and in executing the transactions.

TO HEDGE OR NOT TO HEDGE

The Costs and Benefits of Corporate Hedging A hedged business has fewer risks to worry about, but exactly why is this an advantage? Aren’t businesses supposed to take risks? We answer these questions next, and although we focus on the use of futures, many of these same benefits can be obtained with other derivatives such as forwards, options, and swaps: 1. Hedging locks in a future price. Recall Chapter 8, in which we discussed the establishment of the Chicago Board of Trade. The CBOT built orderly markets for grain trading to avoid price crashes at harvest times and booms afterward. Hedging with futures helps smooth such demand–supply imbalances. It allows traders to lock in stable prices and plan production and marketing activities with greater certainty.

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2. Hedging permits forward pricing of products. For example, many airlines routinely hedge aviation fuel prices, which is a major input cost (see Extension 13.1 for a discussion). If fuel costs can be fixed (and other costs, such as staff salary, airplane depreciation, airport gate rental charges, and travel agents’ commissions, can be estimated in advance), then the airline can better set a profit margin and determine future seat prices. Customers like knowing travel costs far in advance. 3. Hedging reduces the risk of default and financial distress. Default means a failure to pay a contractual payment, such as debt, when promised. Default triggers unfavorable outcomes such as lawyer fees, potential bankruptcy costs and liquidation fees, losing control over the company’s assets, and increased costs of doing business because suppliers fear that they may not be paid and customers may worry about quality and service. These are called financial distress costs. A company can reduce the likelihood of incurring these costs by hedging some of the risks it faces. 4. Hedging facilitates raising capital. Because of the decreased risk of default, bankers allow hedgers to borrow a larger percentage of a commodity’s value at a lower interest rate than nonhedgers. This same logic applies to firms. 5. Hedging enables value-enhancing investments. Froot et al. (1994) argue that if external sources of funds (like stock and bond issuances) are costlier to corporations than internally generated funds, then hedging can help stabilize internal cash flows and make them available for attractive investment opportunities. 6. Hedging reduces taxes. A hedge may be used to capture the benefits of a tax loss or take advantage of a tax credit. The basic intuition is simple. Because one can use a tax loss or a credit only when the company has positive profits, it may be worthwhile to smooth out profits by hedging rather than letting them fluctuate. See Extension 13.2 for an example that illustrates this notion. However, hedging also incurs costs. When trading with a speculator, a hedger often pays an implicit fee by trading at a price inferior to the expected future payoff. Of course, brokers must also be paid. The presence of these trading costs constitutes a basic argument against futures hedging. Moreover, businesses must allocate valuable personnel to devise hedging strategies, set up adequate checks and

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balances to prevent rogue employees from ruinously speculating with the firm’s money, ensure compliance with the laws of the land, and meet the government’s accounting requirements in this regard.1 Hedging is analogous to purchasing an insurance policy on some commodity’s spot price. It pays off when spot prices move in an adverse direction. However, as with all insurance policies, there is a cost—the premium. If the spot price does not move adversely, one pays for insurance not used. Risk-averse individuals buy insurance despite this cost, and analogously, firms often hedge. To hedge or not to hedge remains an unsettled question that must be resolved on a case-by-case basis. One often hears any number of catchy maxims: “no risk, no gain” alongside “risk not thy whole wad.” Which to choose requires expertise, finesse, and knowledge, which are acquired through study and experience.

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EXTENSION 13.1: Airlines and Fuel Price Risk Fuel costs can hit the airlines hard. Fuel is a major cost of the airline business; it can be between 10 percent (in good times) and more than 35 percent (in bad times) of their average expenses.2 “When the going gets tough, the tough get going” goes a familiar saying. Crude oil prices had a huge run-up in 2008 and peaked at over $145 per barrel during July of that year. As the oil price went up, the airlines adopted a wide range of measures, some prudent and others drastic, to reduce fuel consumption. Most of these approaches aimed at lowering the aircraft’s weight because a lighter plane is cheaper to fly. Airlines started charging for checked-in luggage, which “killed two birds with one stone”: raised money and reduced weight. Other steps included flying more fuel-efficient planes and reducing the time that plane engines stay on. Some started trading commodities. Yet, crude oil prices started falling and jet fuel prices also declined. In November 2008, crude prices closed below $60 per barrel. A report in the New York Times entitled “Airlines find it difficult, and perhaps unwise, to hedge fuel prices” dated November 13, 2008 reported that the German airline Lufthansa’s “hedging for the current year had been cut to 72 percent from 85 percent of its total fuel bill ... Its hedging for 2009 is down to 57 percent, at an average of $91 a barrel.” Stunned by unprecedented oil price volatility, several other airlines were left holding costly hedges. Many commercial airlines as well as companies like FedEx (which uses its extensive fleet to quickly deliver packages to numerous locations around the globe) use derivatives on oil to hedge. The hedged amount varies. Sometimes they hedge little or none of their exposure; at other times they hedge much more (see Table 13.1 for a description of the fuel hedging at some of the world’s largest airlines based on thier annual reports). Southwest Airlines has a history of running a successful fuel “hedging program.” It has maintained its long track record of profitability into the new millennium by extensive hedging even at a time when many other airlines were reeling from losses and operating under bankruptcy protection. Consider the 2016 annual report of Southwest Airlines, which states “Because the Company uses a variety of different derivative instruments at different price points, the Company is subject to the risk that the fuel derivatives it uses will not provide adequate protection against

1 Barings Bank, one of England’s oldest and most prestigious merchant banks, founded in 1762 by Sir Francis Baring, collapsed in 1995 after a rogue employee, Nick Leeson, lost $1.4 billion of company money speculating in futures contracts. Nowadays, most companies have stronger oversight of the firm’s overall risk situation through the office of a chief risk management officer, who often reports directly to the chief executive officer.

TO HEDGE OR NOT TO HEDGE

261

significant increases in fuel prices and could in fact result in hedging losses, and the Company effectively paying higher than market prices for fuel, thus creating additional volatility in the Company’s earnings. … Excluding the impact of hedging, Fuel and oil expense would have decreased by $535 million, or 15.9 percent, compared with 2015, due to lower market jet fuel prices.”

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TABLE 13.1: Fuel “Hedging Program‘ at some of the World’s Largest Airline Companies by Revenue (in US dollars) Airlines

Hedging Policy (as of December 2016 or for the year ending December 31, 2016)

American Airlines Group (USA)

No fuel hedges as of December 31, 2016; fully exposed to fluctuations in fuel prices.

Delta Air Lines (USA)

Manages fuel price risk through a “hedging program” with options, swaps and futures on commodities including crude oil, diesel fuel and jet fuel. Recorded fuel hedge losses of $366 million in 2016 and $741 million in 2015.

United Continental Holdings (USA)

The Company historically hedges a portion of its planned fuel requirements. No outstanding fuel hedges as of December 31, 2016. Although the Company’s current strategy is to not enter into transactions to hedge its fuel consumption, it regularly reviews its strategy based on market conditions and other factors.

Deutsche Lufthansa (Germany)

Fuel costs included a loss due to hedging equal to EUR 905m (One Euro was $1.05 on December 31, 2016). Fuel expenses were about EUR 4.9 billion in 2016. The Lufthansa Group hedges fuel prices with a time horizon of up to 24 months.

Air France-KLM (France)

Based on the forward curve at December 31, 2015, a hedging strategy, approved by the Board of Directors, sets the hedge horizon at two years (a rolling 24 months) and the target hedge ratio at 60%. The hedging uses simple futures or option-based instruments.

International Airlines (United Kingdom)

Fuel price risk is partially hedged through the purchase of oil derivatives in forward markets. The current strategy, as approved by the IAG Management Committee, is to hedge a proportion of the anticipated fuel consumption for the next eight quarters. The company primarily trades forward crude, gas oil and jet kerosene derivative contracts for hedging purposes.

Southwest Airlines (USA)

Discussed in text.

Southwest’s “hedging” experience demonstrates the difficulties in distinguishing hedging from speculation and illustrates that even well-considered actions do not always deliver favorable outcomes. 2

As it is extremely difficult to predict oil prices, it is equally difficult to devise an airline’s fuel hedging strategy, which many airlines refer to as a “hedging program.”

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EXTENSION 13.2: A Hedged Firm Capturing a Tax Loss Hedging with derivatives expands a company’s choices and may allow it to take advantage of tax situations. The next example shows how futures contracts can be used to stabilize earnings so that a company can utilize past losses to reduce current taxes.

EXT. 13.2 EX. 1: Hedging to Capture a Tax Loss ■

Goldmines Inc. (fictitious name) mines, refines, and sells gold in the world market. The company’s profits move in tandem with gold price movements. Assume that the pretax profit is $100 million when gold prices go up (which happens with a 50 percent chance) and 0 if gold prices go down (which also happens with a 50 percent chance). Alternatively, Goldmines can hedge with gold futures and have a known profit of $48 million.

■

Ext. 13.2 Fig. 1 shows these payoffs. We label them as event 1 (profit $100) and event 2 (profit $0). These events are shown in a binomial tree, where the payoff $100 is placed on the upper branch of the tree, while $0 is placed on the lower branch.

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Expected Profit ■

The expected profit = [(Probability of high gold price) × (Payoff $100)] + [(Probability of low gold price) × (Payoff $0)] = 0.50 × $100 + 0.50 × 0 = $50 for the unhedged company.

■

We can show the superiority of the hedging strategy by computing the expected after-tax profits for the unhedged and hedged firm under different scenarios.

After-Tax Expected Profits (Simple Case) ■

Assume that the tax rate is 30 percent. When the gold price is high, $100 million pretax profit gives an after-tax profit of 100 × (1 – tax rate) = $70 million. When the gold price is low, the pretax profit is 0, and so is the after-tax profit. After-tax profits are shown in Ext. 13.2 Fig. 1. Expected after-tax profit for unhedged firm = 0.50 × 70 + 0.50 × 0 = $35 million

■

The hedged firm makes a profit of $48 million irrespective of the gold price: After − tax profit for hedged firm = 48 × 0.7 = $33.6 million

TO HEDGE OR NOT TO HEDGE

■

263

The company can choose either. The actual choice depends on the risk preferences of the company’s management and shareholders.

After-Tax Expected Profits (When $25 Million Tax-Deductible Loss Is Carried Forward) ■

Now suppose that the company has accumulated losses totaling $25 million. It can deduct this loss from this year’s profit and thus lower the tax burden. Such strategies of tax reduction are known as tax shields.

■

Assume that if unutilized, this one-shot opportunity disappears. We will show that the hedged firm can always utilize the losses to lower its taxes, but the unhedged firm can only do this half the time.

■

For the unhedged firm, total tax when the gold price is high is (Pretax profit − Loss deducted) × (Tax rate) = (100 − 25) × 0.3 = $22.5 million

EXT. 13.2 FIG. 1: Profits of a Hedged and an Unhedged Firm Pre-Tax Profit 0.5

$100.00 Hedged Firm

Unhedged Firm

$48

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0.5

0 Expected profit of the unhedged firm = 0.5 × 100 + 0.5 × 0 = $50 million Expected After-Tax Profit (with No Loss Carried Forward) 0.5

$70.00

Unhedged Firm

Hedged Firm

$33.60

0.5

0 Expected profit = $35 million Expected After-Tax Profit (with $25 Million Tax Deductible Loss) 0.5

$77.50

Unhedged Firm

Hedged Firm 0.5

0 Expected profit = $38.75 million

$41.10

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The total tax when the gold price is low is 0. Consequently, the after-tax profit is (100 – 22.5) = $77.5 million when the gold price is high and zero when it is low (see Ext. 13.2 Fig. 1): Expected after-tax profit for the unhedged firm = 0.50 × 77.5 + 0.50 × 0 = $38.75 million ■

By contrast, the total tax for the hedged firm is (48 – 25) × 0.3 = $6.90: After-tax profit for the hedged firm = (48 – 6.90) = $41.10 million

■

Clearly hedging is a superior strategy: not only does the hedged firm generate greater after-tax profit but it also removes swings and stabilizes this amount.

13.3

Hedging with Futures

Chapter 4 gave us an inkling of how a company can use forward contracts for hedging input and output price risks. Now that you are more familiar with the workings of a futures contract (which are easier to use than forward contracts), let us explore hedging with futures.

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Perfect and Imperfect Hedges Ours is an imperfect world, in which it is hard to find perfect hedges. A perfect hedge completely eliminates spot price risk for some commodity. This happens when (1) the futures is written on the commodity being hedged, (2) the contract matures when you are planning to lift the hedge, and (3) the contract size and the other characteristics of a futures unerringly fit the hedger’s need. Imperfect or cross hedges occur when these three conditions are not satisfied. For example, a bank may reduce the price risk of its loan portfolio, which is vulnerable to interest rate changes, by trading Treasury bond futures contracts. This is a cross hedge because the futures is written on US Treasury bonds, whereas the loans being hedged consist of, say, house mortgages, car loans, and certificates of deposit. Even when futures on the same commodity are available, the issue of a timing mismatch may occur. Consequently, basis risk emerges as a paramount concept in analyzing, setting, and managing a futures hedge.

Basis Risk Basis risk is a focal point in understanding futures hedges. The basis is defined as the difference between the spot and the futures price. It is written as Basis = Cash price – Futures price. Example 13.1 shows how crucial basis risk is when hedging a

HEDGING WITH FUTURES

commodity’s spot price risk. We illustrate this in the context of two companies using buying and selling hedges to offset input and output risks, respectively.

EXAMPLE 13.1: A Gold Futures Basis Risk ■

■

Today is January 1, which is time 0. Consider the gold futures contract trading on the COMEX Division of the CME Group described in Chapter 8. The prices are reported on a per ounce. Figure 13.2 gives the timeline and the various prices. For simplicity, we assume that no interest is earned on margin account balances. On today’s date, the spot price of gold S = $990, and the June contract’s futures price F(0) = $1,000. The basis is b (0) = Spot price S (0) − Futures price F (0) = 990 − 1,000 = −$10

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FIGURE 13.2: Timeline and Prices for Gold Futures Hedging Example

■

Start date (January 1) Time 0

Closing date (May 15) Time T

Delivery period (June)

Spot price, S(0) = $990 Futures price, F (0) = $1,000 Basis, b(0) = S(0) – F (0) = –$10

S(T) = $950 F (T) = $952 b(T) = S(T) – f (T) = –$2

The delivery period for the contract is in June. For the subsequent discussion, suppose that on May 15, time T, the spot price S(T) is $950 and the June futures price F(T) is $952. The new basis is b (T) = S (T) − F (T) = 950 − 952 = −$2

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Goldmines Inc. Sets Up a Short Hedge ■

■

Suppose that the mining company Goldmines Inc. goes short one June gold futures contract on January 1 to hedge its output price risk. The company sells gold and lifts the hedge by closing out the futures position on May 15. Goldmines sells gold for $950, but it makes – [F(T) – F(0)] = – (952 – 1,000) = $48 on the futures position. So, the effective selling price on May 15 is (see Equation 9.1b of Chapter 9) Spot cash flow + Futures cash flow = 950 + 48 = $998

■

One can rewrite this as Spot − (Change in futures price) = S (T) − [F (T) − F (0)] = [S (T) − F (T)] + F (0) = New basis + Old futures price

(13.1)

No matter when the company closes its position, the old futures price is fixed, and only the new basis affects profits.

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Jewelrygold Inc. Sets Up a Long Hedge ■

Now look at the same problem from the perspective of a jewelry maker, Jewelrygold Inc. As discussed in chapter 4, the company can reduce input price risk by setting up a long hedge (a buying hedge) and going long gold futures. When it is time to buy gold in the spot market, the company sells these futures and removes the hedge.

■

On January 1, Jewelrygold goes long a June gold futures contract to hedge input price risk. On May 15, the company buys gold for $950 and simultaneously sells futures for $952 to end the hedge. Although Jewelrygold pays less for gold, it has to surrender nearly all of its gains through the hedge. The cash flow from the futures position is [F(T) – F(0)] = 952 – 1,000 = –$48.

■

Jewelrygold’s effective price on May 15 is $998. By looking at the cash flows, the buying price is = −Spot + (Change in futures price) = −S (T) − [F (T) − F (0)] = − [S (T) − F (T) + F (0)] = − (New basis + Old futures price)

(13.2)

This is the same value as in the previous example, except for the minus sign.

This example shows that a futures hedged spot commodity’s portfolio value can always be viewed as the sum of the old futures price and the new basis. Consequently, hedgers are interested in how the basis evolves randomly through time. This randomness gives rise to basis risk in hedging, which is often measured by computing the basis’s variance (or standard deviation). Basis risk is, thus, fundamental

HEDGING WITH FUTURES

to futures hedging, and hedgers often talk about the widening or narrowing of the basis in passionate terms. Many of them have extensive charts depicting the historic behavior of the basis, looking for a crystal ball to foretell the future.

Guidelines for Futures Hedging Real life is far more complex than textbook examples. Exchanges offer only a handful of futures, whereas there are thousands of commodities. The specter of an imperfect hedge visits us again. When setting up a hedge, it’s natural to ask, which contract and for what maturity? Example 13.2 discusses the issues involved in answering this question.

EXAMPLE 13.2: Hedging by Selecting a Futures from Several Competing Contracts Suppose that a new alloy Alloyum (a fictitious name) is developed that can replace precious metals in industrial and ornamental use. Let Alloyum trade in the spot market. Alloyum futures may or may not trade, and we discuss both possibilities. If you produce thirty thousand ounces of Alloyum, how would you hedge output price risk?

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Alloyum Futures Trade, and You Know When to Lift the Hedge This case happens when the Alloyum futures contract’s maturity date exactly matches the delivery date of the spot commodity commitment. Then, sell [(Spot position)/(Alloyum futures contract size)] = 30,000/50 = 600 Alloyum futures, assuming a contract size of fifty ounces. Basis risk eventually disappears because the spot converges to the futures price at maturity. This can be shown by setting S(T) = F(T) in Equation 13.1. With zero basis risk, you fix the selling price at maturity, which is none other than today’s futures price. Congratulations, you have set up a perfect hedge. (This perfect hedge ignores the interest rate risk from marking-to-market. Such reinvestment risks will be considered later in the chapter.)

Alloyum Futures Trade, and You Do Not Know When to Lift the Hedge In this case, one way to proceed is to compute the variance of the basis for different maturity futures contracts on Alloyum and select the one that scores the lowest. Lower basis risk means that the futures price deviates less from the spot price. You are likely to find that the futures contract maturing in the same month as the spot sale is the best candidate. In this case, hedge with the smallest basis risk futures contract. The number of contracts to be shorted may be found by the risk-minimization hedging strategy discussed later in this chapter.

Alloyum Futures Do Not Trade, and You Know When to Lift the Hedge In this case, you may first like to select a similar commodity on which a futures contract is written and decide on the maturity month. Collect price data for Alloyum and for contender commodity futures contracts maturing in the same month as the spot sale, and compute correlations of price changes. Select the commodity futures that has the highest correlation to Alloyum in the overall period.

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Alloyum Futures Do Not Trade, and You Do Not Know When to Lift the Hedge This is the worst case for hedging. First, select a commodity futures contract as in the previous case. Then, select a maturity month, and find the futures contract with the largest correlation with Alloyum spot price changes and that minimizes the variance of the basis. There is, however, a famous saying in economics that “there are no solutions, only trade-offs” (Sowell 1995, p 113). The other futures contracts may have lower transaction costs. Liquidity is another concern because it often increases as the contract approaches maturity, but dries up in the delivery month.

13.4

Risk-Minimization Hedging

Suppose that you want to hedge a long spot commodity position by shorting futures contracts. Using a statistical or econometric model, assuming that past price patterns repeat themselves enables you to find the optimal number of contracts to minimize the variance of the hedging error. This section discusses this statistical approach to hedging.

The Mean-Variance Approach The mean-variance approach to risk-minimization hedging (or optimal hedging) determines the optimal number of contracts needed to minimize the variance of the portfolio’s price changes. The intuition of this approach can be understood by first considering a perfect hedge: Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

A perfect hedge. Suppose you are holding a long position in the spot commodity. You need to short futures contracts to hedge the spot commodity’s price risk. In the pristine world of a perfect hedge, any change in the spot position will be exactly offset by an equal and opposite change in the futures position. Consequently, Change in portfolio value = Change in spot position – Change in futures position = 0

(13.3)

The minus sign before the change in the futures position is because we are short the futures contract and the spot and futures prices move in the same direction. With minor modifications, we can recast the problem to focus on how changes in the spot and futures prices per unit are related. Using data from Example 13.2, we can hedge a long position of thirty thousand ounces of Alloyum production by selling [(Cash position)/(Alloyum futures contract size)] = 30,000/50 = 600 Alloyum futures. We can write this as follows: Change in portfolio value = (30, 000 × Change in spot price per ounce) − (600 × 50 × Change in futures price per ounce) =0 (13.4)

RISK-MINIMIZATION HEDGING

There is no change in the portfolio value because the spot and the futures price changes exactly match. ■

An imperfect hedge. In reality, the hedge will be imperfect because the spot and the futures price changes do not exactly match. Here you can find the risk-minimizing number of futures contracts. Consider hedging our spot exposure of n ounces by selling q futures contracts of size f ounces per contract. Then we can rewrite the left side of Equation 13.4 as Change in portfolio value = (n × Change in spot price per ounce) − (q × f × Change in futures prices per ounce) = n [S (t) − S] − qf [F (t) − F] (13.5) = nΔS − qfΔF where S(t) and S are the spot prices per ounce on some future date (time t) and today (time 0), F(t) and F are futures prices per unit on those same respective dates, and Δ compactly denotes the price change. Note that this change does not equal zero because it is an imperfect hedge. To find the number of contracts to sell to risk-minimize hedge the spot portfolio, compute the variance of the portfolio and select n to minimize this variance. You can do this by taking the partial derivative of the portfolio variance with respect to q, setting it equal to zero, and solving for q (see the appendix to this chapter). This leads to the following result (see Result 13.1).

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RESULT 13.1 The Risk-Minimizing Number of Futures Contracts for Hedging a Spot Commodity Position To minimize the risk of a long portfolio of a spot commodity, sell short q contracts where q = (n/f) h (13.6) and n is the size of the spot position, f is the number of units of the underlying commodity in one futures contract, and h is the optimal hedge ratio (or minimum-variance hedge ratio or risk-minimized hedge ratio). The optimal hedge ratio is given by h=

covS,F sdS = corr varF sdF S,F

(13.7)

where covS,F is the covariance between changes in the spot price (ΔS) and the futures price (ΔF), varF is the variance of the change in the futures price, sdS is the standard deviation (or volatility) of the change in the spot price (standard deviation is the positive square root of the variance), sdF is the

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standard deviation of change in the futures price, and corrS,F is the correlation coefficient between ΔS and ΔF.c c

The second part of the formula follows from the result: “covariance equals the product of the standard deviations and the correlation coefficient” or covS,F = sdS × sdF × corrS,F .

Considering various cases reveals the intuition behind this result. Suppose that the price changes for the spot commodity and the futures contract are perfectly correlated (i.e., corrS,F equals 1) and their standard deviations match (i.e., sdS = sdF ). Then the optimal hedge ratio h equals 1. This is the holy grail of a perfect hedge, which is nearly impossible to attain in practice but is useful as an illustration. Next assume that there is a high correlation between the price changes (corrS,F is close to one) and the spot price change has a lower volatility than the futures price change (i.e., sdS < sdF ). Then a unit of spot is hedged with less than a unit of futures. Conversely, if the spot price change fluctuates more than the futures price change (i.e., sdS > sdF ), then one unit of spot needs more than one unit of the futures to hedge. The optimal hedge ratio gives us an easy formula for setting up a futures hedge. Example 13.3 shows how to implement this by utilizing spot and futures price data, which can be easily collected from business newspapers or Internet data sources.

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EXAMPLE 13.3: Setting Up a Risk-Minimization Hedge with Price Data ■

Suppose that you are planning to sell thirty thousand ounces of Alloyum at some future date. To start, you collect Alloyum spot price and platinum futures price data for sixteen consecutive trading days (see Table 13.2).

■

Begin by computing the price differences for each of these two series of price observations. Next use a spreadsheet to compute the parameter estimates needed for the optimal hedge ratio. (The appendix to this chapter discusses related issues and demonstrates how to find h with the help of a simple calculator; Section 13.6 shows how to do these calculations using the spreadsheet program Microsoft Excel.) We get

■

the standard deviation of changes in the futures price per unit, sdF = 14.3368 the standard deviation of changes in the spot price sdS = 15.5002 the correlation coefficient between the spot and futures price changes, corrS,F = 0.7413 Use these estimates in the second part of Equation 13.7 of Result 13.1 to compute the minimumvariance hedge ratio: sdS 15.5002 h= corrS,F = 0.7413 = 0.80154 14.3368 sdF

RISK-MINIMIZATION HEDGING

To hedge n = 30,000 ounces of Alloyum with a platinum futures contract (contract size f = 50), you need to sell

■

q = (n/f) h = (30, 000/50) 0.8015 = 480.9 or 481 contracts (with integer rounding) 4 Or, you can use the first part of Result 13.1 to determine the optimal hedge ratio, h = cov /var = 18.3048/22.8381 = S,F F 0.8015.

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TABLE 13.2: Alloyum Spot and Platinum Futures Price Data Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,215

1,233

1

1,209

1,236

2

1,239

1,245

3

1,245

1,254

4

1,254

1,272

5

1,227

1,254

6

1,230

1,248

7

1,224

1,242

8

1,239

1,260

9

1,251

1,263

10

1,227

1,254

11

1,215

1,227

12

1,209

1,221

13

1,203

1,230

14

1,185

1,203

15

1,194

1,209

Alternatively, you can compute the hedge ratio by suitably framing the problem and estimating a linear regression (see the appendix to this chapter).

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Limitations of Risk-Minimization Hedging Comedian Rodney Dangerfield often chuckled, “I don’t get no respect.” Doesn’t this comment apply to risk-minimization hedging? It is purely computational, and except for some ad hoc adjustments, it is a single-shot exercise with no prescription for modifying the hedge over time. It works well if a closely related commodity can be found and you know when to lift the hedge. It works less well otherwise and if the hedge needs to be rolled over or rebalanced. The idea of rebalancing is related to the concept of (exact) dynamic hedging, which involves regular adjustments to a perfect hedge over time. Dynamic hedging requires sophisticated analytical tools, and it applies in a complete market. It is impossible to implement in an incomplete market (see Duffie [1989] and Jarrow and Turnbull [2000] for examples and issues related to dynamic hedging). In complete markets (remember Chapter 8), all kinds of securities trade that generate future payoffs contingent on all possible future events. In reality, the markets are incomplete, and it may be difficult to develop good futures pricing models to which dynamic hedging applies. Consequently, risk-minimization hedging, despite all its limitations, starts looking respectable again. In incomplete markets, it continues to attract serious research interest.

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13.5

Futures versus Forward Hedging

It’s common practice in many circles to treat forwards and futures as equivalent and interchangeable contracts. However, this is incorrect for many reasons. Although a forward is easy to price, ordinary investors rarely trade in the “organized” forward market. Big banks, large corporations, and other institutions with excellent credit ratings dominate this market, and to guarantee execution of the contracts, these financial institutions usually need to post collateral. These trading restrictions are imposed to reduce “counterparty credit risk”—the nonexecution of the contract terms. Counterparty credit risk is a big concern in the trading of over-the-counter derivatives, as recently evidenced by the related regulatory reforms following the 2007 credit crisis in the Dodd–Frank Wall Street Reform and Consumer Protection Act. Second, given that forward contracts are bilateral negotiated agreements, forward markets are illiquid and subject to significant liquidity risk. For example, if a counterparty closes a forward contract early, significant closing costs are incurred. By contrast, a futures (1) is an exchange-traded, standardized contract; (2) has margins and daily settlement, which makes it safer than a forward due to the absence of credit risk; (3) allows small traders, weaker credits, and complete strangers to participate; and (4) usually trades in a liquid market, where traders can enter or unwind their positions with ease. But there is another important difference between futures and forward contracts due to the marking-to-market of a futures contract. Marking-to-market a futures contract introduces risks involved with reinvesting the cash flows before the contract matures. These same reinvestment risks are not faced by a forward contract. Tracking

SPREADSHEET APPLICATIONS: COMPUTING h

the cash flows to these contracts, the futures trader, unlike the forward trader, earns random and uncertain interest on margin balances. As discussed in Chapter 9, the interest earned on a futures margin account is dependent on the futures price changes. In addition, interest rate changes also affect the interest earnings. Indeed, if interest rate changes are positively correlated with futures price changes, then the futures position benefits from interest rate movements because as cash flows are received, more interest is earned. In this case, the risk of a futures position is reduced slightly. Conversely, if interest rate changes are negatively correlated with futures price changes, then the futures position suffers from interest rate movements. In this latter case, the risk is increased. This interest rate risk affects hedging performance because of the random cash flows received. This same risk is not present in a forward contract.

13.6

Spreadsheet Applications: Computing h

You can easily compute the optimal hedge ratio (h) by using a standard spreadsheet program such as Microsoft Excel. Next, we demonstrate this for Example 13.4.

EXAMPLE 13.4: Computing the Optimal Hedge Ratio (Solved Using Microsoft Excel)

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Generating the Inputs Consider the same data as in Example 13.3. Type in the numbers and the terms exactly as follows in an Excel spreadsheet (see Table 13.4): ■

Type Day t in cell A1, 0 in A2, 1 in A3, 2 in A4, and so on up to 15 in cell A17. You may speed up data entry by using the Auto Fill feature: type 0 in A2 and 1 in A3, highlight the cells A2 and A3, and then drag down the bottom right hand corner of the cursor so that the desired values get filled in cells A4 to A17.

■

Type S(t) in B1 and then fill out the Alloyum spot prices as given in Table 13.2: 1,215 in B2, 1,209 in B3, and so on, up to 1,194 in B17.

■

Type F(t) in C1 and then fill out the platinum futures prices F(t) as given in Table 13.2 for 16 consecutive trading days.

■

Type ΔS(t) in D1 and then type “= B3–B2” in D3, “= B4–B3” in D4, and so on. You may speed this up by using the Auto Fill feature: type “= B3–B2” in D3 and then drag down the bottom right hand corner of the cursor so that the desired values fill in cells D4 to D17.

■

Type ΔF(t) in E1 and then type “= C3–C2” in E3. Use Auto Fill as we have just mentioned and fill out the desired values in cells E4 to E17.

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TABLE 13.3: Computing h Using Microsoft Excel A

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Day t

B

C

D

E

S(t)

F(t)

ΔS(t)

ΔF(t)

F

G

H

ΔS(t) × ΔS(t) ΔF(t) ×ΔF(t)

ΔS(t) ×ΔF(t)

0

1,215 1,233

1

1,209 1,236

–6

3

36

9

–18

2

1,239 1,245

30

9

900

81

270

3

1,245 1,254

6

9

36

81

54

4

1,254 1,272

9

18

81

324

162

5

1,227 1,254

–27

–18

729

324

486

6

1,230 1,248

3

–6

9

36

–18

7

1,224 1,242

–6

–6

36

36

36

8

1,239 1,260

15

18

225

324

270

9

1,251 1,263

12

3

144

9

36

10

1,227 1,254

–24

–9

576

81

216

11

1,215 1,227

–12

–27

144

729

324

12

1,209 1,221

–6

–6

36

36

36

13

1,203 1,230

–6

9

36

81

–54

14

1,185 1,203

–18

–27

324

729

486

15

1,194 1,209

9

6

81

36

54

Sums

–21

–24

3,393

2,916

2,340

VARA

240.2571

205.5429

STDEV

15.5002

14.3368

COVAR

164.7429

CORREL

0.74138

h

0.8015

0.8015

SUMMARY

Computing Estimates from the Sample and the Minimum-Variance Hedge Ratio The entries in columns D and E are the two data series with which we work. You can do the computations as follows (and verify that these are the same as the sample variances that you computed in Example 13.3): ■

Type in a name for the sample variance such as vara in cell A20, type “= vara(d3:d17)” in D20 and hit return to get 240.2571, and type “= vara(e3:e17)” in E20 and hit return to get 205.5429.

■

Type stdev in A21, type “= stdev(D3:D17)” in D21 and hit return to get 15.50023, and type “= stdev(e3:e17)” in E21 and hit return to get 14.33677.

■

As Excel’s covariance formula is given for the population, you need to multiply the estimator by n/(n – 1) to get the sample estimate. Accordingly, type covar in A22, and type “= COVAR(D3:D17,E3:E17)*(15/14)” in D22 and hit return to get 164.7429.

■

Type correl in A23 and “= correl(d3:d17,e3:e17)” in D23 and hit return to get 0.74134.

■

Type h in A24 and “= d22/e20” in D24 and hit return to get 0.801501. This is the Equation 13.6 given in Result 13.1. You may also get h via the second result in Equation 13.6 by typing “= d23*d21/e21” in E24 and hitting return—it’s the same answer.

13.7

Summary

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1. Most futures contracts are traded to hedge spot commodity price risk. Producers may set up a long hedge (or a buying hedge) to fix the buying price for an input and a short hedge (or a selling hedge) to fix the selling price of an output. 2. As with any other derivatives, the costs and the benefits must be carefully weighed before hedging a spot commodity with futures. They involve direct and indirect costs: the brokers must be paid, and a trade may fetch a bad price. But hedging also has many potential benefits: it can stretch the marketing period, protect inventory value, permit forward pricing of products, reduce the risk of default and financial distress costs, and perhaps facilitate taking advantage of a tax loss or tax credit. 3. A perfect hedge completely eliminates price risk. It happens when (1) the futures is written on the commodity being hedged and (2) the contract matures when you are planning to lift the hedge. Generally, there is basis risk. The basis is defined as the difference between the spot price and the futures price. 4. For a seller of a futures contract who hedges output price risk, the effective selling price is (new basis + old futures price). For a buyer of a futures contract who hedges input price risk, the effective buying price is the negative of (new basis + old futures price). 5. We can use a statistical model, assuming past price patterns repeat themselves, to set up a risk-minimization hedge. To minimize the risk of a long portfolio of the spot commodity, sell short q = (n/f )h contracts, where n is the size of the spot position, f is the contract size, and h is the optimal hedge ratio (or

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minimum-variance hedge ratio) defined as covS,F /varF , where covS,F is the sample covariance between the change in the spot price (ΔS) and the change in the futures price (ΔF) and varF is the variance of the change in the futures price.

13.8

Appendix

Deriving the Minimum-Variance Hedge Ratio (h) Suppose you are planning to sell n units of a spot commodity at some future date and decide to hedge this exposure by selling short q futures contracts today. ■

Rewrite the change in the portfolio value between today (time 0) and some future date (time t) as ΔPortfolio = (n × Change in spot price) − (q × f × Change in futures price) = n [S (t) − S (0)] − q f [F (t) − F (0)] = nΔS − q fΔF

(13.8)

where f is the number of units of the futures contract, S(t) is the spot price at date t, F(t) is the futures price at date t, and Δ denotes a price change. ■

Select q so as to minimize the change in the portfolio value. First, use statistics to compute the variance of the portfolio’s price change:

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variance (aX − bY) ≡ a2 variance (X) + b2 variance (Y) − 2ab [covariance (X, Y)]

(13.9)

where a and b are constants and X and Y are random variables. As n, q, and f are constant parameters, set a = n, b = q f, X = ΔS, and Y = ΔF in Equation 13.9 to get 2

var (ΔPortfolio) = n2 vars + (q f) varF − 2nq f × covS,F

(13.10)

where varS is the variance of change in the spot price (ΔS), varF is the variance of change in the futures price (ΔF ), and covS,F is the sample covariance between the change in the spot and futures prices. ■

Using calculus, we minimize Equation 13.10 by taking the partial derivative with respect to q and setting it equal to zero: 𝜕 [var (ΔPortfolio)] = 2q f 2 varF − 2nf covS,F = 0 𝜕q The second partial derivative is a positive number, which indicates that this minimizes the equation. Rearrange terms to get the number of futures contracts to sell short (or go long in case you are setting up a buying hedge): q = (n/f) h where the optimal hedge ratio h equals (covS,F /varF ).

(13.11)

APPENDIX

Computing the Minimum-Variance Hedge Ratio (h) Example 13.4 shows how to input the values into a business calculator or a computer to determine the minimum variance hedge ratio (h). Here we show how to directly calculate h.

Statistical Approach Start by deciding how frequently and over what time interval you collect your data. Standard practice is to fix a time interval (daily, weekly, or monthly) and use the end of the interval’s settlement prices.

DATA SERIES ■

To cross-hedge Alloyum spot with platinum futures, we need their prices as basic inputs. These are reported in Table 13.5, whose first three columns are as follows: - The first column is labeled Day t, and it keeps track of the days. There are (T + 1) = 16 observations corresponding to sixteen consecutive trading days, where t “runs” from day 0 to day t = 15. - Columns 2 and 3 record the Alloyum spot price S(t) and the platinum futures price F(t), respectively, for these sixteen consecutive trading days. All prices are reported for one unit of the respective commodity.

PRICE DIFFERENCES

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■

Compute price differences for each price series. - Column 4 reports changes in the value for Alloyum spot. Denote the spot price change from date (t – 1) to date t as ΔS(t) ≡ S(t) – S(t – 1). Label column 4 as x(t) for convenience. For example, when t = 6, x (6) ≡ ΔS (6) − S (6) – S (5) = 1, 230 − 1, 227 = 3 - Column 5 does the same for platinum futures. Label this as y(t). For example, y (6) ≡ ΔF (6) = ΔF (t) ≡ F (t) − F (t − 1) = F (6) − F (5) = 1, 248 − 1, 254 = −6

■

Notice that when computing the “difference,” you lose the first observation. We now have T = 15 observations. We denote them by t, where t = 1, 2, . . . , 15.

■

We can now forget our second and third columns. We use the fourth and fifth columns to compute the minimum-variance hedge ratio h.

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TABLE 13.4: Price Data and Calculations for the Minimum-Variance Hedge Ratio (h) ΔS(t) ≡ S(t) – S(t – 1) ≡ x(t)

ΔF(t) ≡ F(t) – F(t – 1) ≡ y(t)

1,236

–6

3

1,239

1,245

30

3

1,245

1,254

4

1,254

5

[ΔF(t)]2 ≡ y(t)2

ΔS(t) × ΔF(t) ≡ x(t)y(t)

36

9

–18

9

900

81

270

6

9

36

81

54

1,272

9

18

81

324

162

1,227

1,254

–27

–18

729

324

486

6

1,230

1,248

3

–6

9

36

–18

7

1,224

1,242

–6

–6

36

36

36

8

1,239

1,260

15

18

225

324

270

9

1,251

1,263

12

3

144

9

36

10

1,227

1,254

–24

–9

576

81

216

11

1,215

1,227

–12

–27

144

729

324

12

1,209

1,221

–6

–6

36

36

36

13

1,203

1,230

–6

9

36

81

–54

14

1,185

1,203

–18

–27

324

729

486

15

1,194

1,209

9

6

81

36

54

Σx = – 21

Σy = – 24

Σx2 = 3,393

Σy2 = 2,916

Σxy = 2,340

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Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,215

1,233

1

1,209

2

Sums

[ΔS(t)]2 ≡ x(t)2

Some values useful for the computations follow: T

∑

x (t) = −21

∑

y (t) = −24

∑

x(t)2 = 3, 393

∑

y2 = 2, 916

∑

x (t) y (t) = 2, 340

t=1 15

t=1 15

t=1 15

t=1 15

t=1

APPENDIX

COMPUTATION-SIMPLIFYING TECHNIQUES - Three more columns are introduced to help simplify our calculations. - Columns 6 and 7 record the square of the price changes reported in Columns 4 and 5, respectively. Finally, Column 7 reports the product of the values in Columns 5 and 6. For example, when t = 6, we get [x(6)]2 = 32 = 9y (6) [ y(6) ] 2 = (−6)2 = 36 x (6) y (6) = −18

STATISTICAL ESTIMATES ■

To compute h from historical data, modify our formulas for covariance and variance by replacing the T with (T - 1) in the denominator to get an unbiased estimate.

■

We get the following parameter estimates to help us compute h (see Table 13.6): sample covariance, covS,F = 164.7429 sample variance of ∆S, varS = 240.2571 sample variance of ∆F, varF = 205.5429 sample standard deviations, sdS = 15.5002 and sdF = 14.3367 correlation coefficient between ΔS and ΔF, corrS,F = 0.7413

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■

These estimates are used to compute the minimum-variance hedge ratio h. The first part of Equation 13.7 of Result 13.1 gives h=

covS,F 164.7429 = = 0.8015 varF 205.5429

(13.12)

while the second part verifies the same result: h= ■

sdS 15.5002 0.7413 = 0.8015 corr = sdF S,F 14.3367

(13.13)

If you are estimating these values over a longer period, then you run the risk that a futures contract may mature and no longer trade. Suppose you are hedging with the “nearest maturity” futures contract, whose price changes are likely to have the best correlation with the spot price changes. Suppose this contract stops trading on Day 12. Select the futures contract that is going to mature next, get its futures prices for Days 12 and 13, and write the price difference ΔF(13) ≡ F(13) – F(12) in the first table in the cell corresponding to Day 13’s price difference.

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CHAPTER 13: FUTURES HEDGING

TABLE 13.6: Statistical Estimates and Computation of the Optimal Hedge Ratio h Statistical Estimate

Formula

Computations

Estimated Value

–21/15

–1.4000

T

a. Sample mean, x

1 x(t) T∑ t =1

b. Sample mean, y

1 y(t) 15 ∑ t =1

–24/15

–1.6000

1 [x(t) − x] × [y(t) − y] T−1 ∑ t =1

1 −21 −24 2, 340 − 15 ( 15 ) ( 15 )] 14 [

164.7429

15

T

c. Sample covariance of ΔS and ΔF, covS,F

15

=

1 x(t)y(t) − 15xy 14 {[∑ } ] t =1 T

d. Sample variance of ΔS, varS

1 [x(t) − x]2 T−1 ∑ t =1

240.2571

15

=

1 2 x(t)2 − 15x 14 [∑ ] t =1 T

2

1 [y(t) − y]2 T−1 ∑ t =1

1 −24 2, 916 − 15 ( 14 [ 15 ) ]

205.5429

f. Sample standard deviation of ΔS, sdS

+√varS

+√240.2571

15.5002

g. Sample standard deviation of Δ F, sdF

+√varF

+√205.5429

14.3367

covS,F

164.7429 15.5002 × 14.3367

0.7413

164.7429 205.5429

0.8015

e. Sample variance of ΔF, varF Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

2

1 −21 3, 393 − 15 ( 14 [ 15 ) ]

h. Correlation coefficient between ΔS and ΔF, corrS,F i. Minimumvariance hedge ratio, h j. h (alternate computation)

sdS × sdF

covS,F varF sdS corrS,F sdF

0.7413

15.5002 ( 14.3367 )

0.8015

CASES

Econometric Approach: A Linear Regression Model Econometrics is the branch of economics that studies relationships between economic variables using mathematics and statistics. A staple of econometrics is the linear regression model,6 in which a dependent variable may be determined from the values of one or more independent variables, under the assumption that these variables are linearly related. In this case, the model takes the form ΔS (t) = 𝛼 + 𝛽ΔF (t) + u (t)

(13.14)

where 𝛼 and 𝛽 are unknown parameters and u(t) are independent and identically distributed error terms with an expected value of 0 and a constant variance. Next you have to estimate these parameters. The most popular approach is the least squares method, in which 𝛼 and 𝛽 are chosen so that the sum of squared errors Σt [ΔS(t) – 𝛼 – 𝛽ΔF(t)]2 is minimized. Interestingly, the value of 𝛽 obtained is identical to the hedge ratio h in Result 13.1. This makes life easy. Estimate the regression Equation 13.4 by hand or by using a standard statistical package like MATHEMATICA, MINITAB, SAS, or TSP, and the estimated value of 𝛽 will give you the hedge ratio. Spreadsheet programs like Microsoft Excel can also run regressions.

13.9

Cases

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Lufthansa: To Hedge or Not to Hedge . . . (Richard Ivey School of Business

Foundation Case 900N22-PDF-ENG-PDF-ENG, Harvard Business Publishing). A short case that explores the costs and benefits of hedging and the derivatives that may be used for hedging purposes. Enron Gas Services (Harvard Business School Case 294076-PDF-ENG). The case

considers the risks and opportunities of selling a variety of natural gas derivatives by a financial services subsidiary of the largest US integrated natural gas firm. Aspen Technology Inc.: Currency Hedging Review (Harvard

Business School Case 296027-PDF-ENG). The case examines how a small, young firm’s business strategy creates currency exposure and how one can manage such risks.

6

See standard econometrics textbooks like Amemiya (1994) or Johnston and DiNardo (1997) for a discussion of the linear regression model.

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13.10 Questions and Problems 13.1. Define a long hedge and a short hedge and give examples of each kind of

hedge. 13.2. What are the benefits of a corporation’s hedging? In your answer, explain

why the corporation and not the corporation’s equity holders must do the hedging. Are there any costs to a corporation’s hedging? 13.3. What is the difference between a perfect hedge and a cross hedge? Give

examples to clarify your answer. 13.4. If one cannot create a perfect hedge, what are the alternatives? Give an

example to explain your answer. 13.5. What is the basis for a futures contract? What happens to the basis on the

delivery date? 13.6. There is no futures contract on aviation fuel. Combo Air Inc. has to buy

3 million gallons of aviation fuel in three months. Suppose you are in charge of Combo Air’s hedging activities. You gather the following data:

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TABLE 13.7: Jet Fuel Cash Prices vs. Near Month Energy Futures Prices: Correlations of Price Changes 1986–88

1986

1987

1988

Heating oil futures

0.54

0.76

0.89

0.32

Gasoline futures

0.41

0.74

0.73

0.19

Crude oil futures

0.45

0.70

0.72

0.25

a. Which energy futures contract will you choose for hedging jet fuel

purchase? b. Is this a long hedge or a short hedge? 13.7. The variance of monthly changes in the spot price of live cattle is (in cents per

pound) 1.5. The variance of monthly changes in the futures price of live cattle for the April contract is 2. The correlation between these two price changes is 0.8. Today is March 11. The beef producer is committed to purchasing four hundred thousand pounds of live cattle on April 15. The producer wants to use the April cattle futures contract to hedge its risk. What strategy should the beef producer follow? (The contract size is forty thousand pounds.)

QUESTIONS AND PROBLEMS

13.8. Are hedging with forwards and futures contracts the same, or are there

different risks to be considered when using these two contracts? Explain your answer. 13.9. When you hedge a commodity’s price risk using a futures contract, give an

example where the counterparty is also hedging. Give an example where the counterparty is speculating. 13.10. Kellogg will buy 2 million bushels of oats in two months. Kellogg finds that

the ratio of the standard deviation of change in spot and futures prices over a two-month period for oats is 0.83 and the coefficient of correlation between the two-month change in price of oats and the two-month change in its futures price is 0.7. a Find the optimal hedge ratio for Kellogg. b How many contracts do they need to hedge their position? (The size of

each oats contract is five thousand bushels; oats trades in the CME Group.) 13.11. Explain why hedging is like buying an insurance policy. To buy an insurance

policy, you need to pay a premium; what is the corresponding premium in hedging? Give an example to clarify your answer. 13.12. Suppose that after you graduate, you plan to be a stock analyst for a major

financial institution. You know that if the stock market increases in value, you will get a job with a good salary. If the stock market declines, you will get a job, but the salary will be lower. How can you hedge your salary risk using futures contracts? Is this a perfect or imperfect hedge? Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

13.13. Your company will buy tungsten for making electric light filaments in the

next three to six months. Suppose there are no futures on tungsten. How would you hedge this risk? (Discuss the type of hedge, general hedging approach, and guidelines that you would like to follow.) 13.14. You are the owner of a car rental business. If gasoline prices increase, your

car rental revenues will decline. How can you hedge your car rental revenue risk using futures contracts? Is this a perfect or imperfect hedge? 13.15. What commodity price risk does Southwest Airlines hedge, and why? Has it

always been successful in its hedging program? 13.16. What is risk-minimizing hedging? Briefly outline how you would set up a

risk-minimizing hedge. Is a risk-minimizing hedge a perfect or imperfect hedge? Explain your answer.

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13.17. Canadian American Gold Inc. (CAG) has half its gold production from

mines located in Canada, while the other half is from those in the United States. CAG uses a quarter of its production for making gold jewelry sold at a fixed price through stores in the two nations, and the rest is sold on the world market, where the gold price is determined in US dollars. Canadian profits are repatriated to the United States, where CAG’s headquarters are located. CAG’s chief executive officer wants to use futures contracts to hedge the entire production of ten thousand ounces of gold and as many other transactions as possible. He communicates his desire to you but seeks your opinion one last time before the orders go out. Devise a sensible hedging strategy that would still be in line with the CEO’s wishes (assume x is the quantity used for making gold jewelry in the United States). 13.18. The spot price of gold today is $1,505 per ounce, and the futures price for a

contract maturing in seven months is $1,548 per ounce. Suppose CAG puts on a futures hedge today and lifts the hedge after five months. What is the futures price five months from now? Assume a zero basis in your answer. 13.19. Suppose that Jewelry Company is planning to sell twenty thousand ounces of

platinum at some future date. The standard deviation of changes in the futures price per ounce sdF is 12.86, that for changes in the spot price per ounce sdS is 14.38, and the correlation coefficient between the spot and futures price changes corrS,F is 0.80. a. Compute the optimal hedge ratio for Jewelry Co. b. How many contracts do they need to hedge their position? (The contract Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

size is fifty ounces.) c. Will this be a buying or a selling hedge?

QUESTIONS AND PROBLEMS

13.20. (Microsoft Excel) Given the following data, compute the hedge ratio for a

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risk minimizing hedge. Day t

Alloyum Spot S(t)

Platinum Futures F(t)

0

1,233

1,245

1

1,219

1,256

2

1,118

1,130

3

1,246

1,264

4

1,250

1,280

5

1,219

1,223

6

1,230

1,248

7

1,227

1,280

8

1,249

1,260

9

1,225

1,289

10

1,227

1,254

11

1,223

1,255

12

1,211

1,223

13

1,203

1,267

14

1,189

1,213

15

1,199

1,219

285

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III

Options

CHAPTER 14

Options Markets and Trading

CHAPTER 15

Option Trading Strategies

CHAPTER 16

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Option Relations

CHAPTER 19

The Black–Scholes– Merton Model

CHAPTER 20

Using the Black– Scholes–Merton Model

CHAPTER 17

Single-Period Binomial Model

CHAPTER 18

Multiperiod Binomial Model

14 Options Markets and Trading 14.1 Introduction Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

14.2 Exchange-Traded Options 14.3 A History of Options Early Trading of Options The Year 1973: The Watershed Year and After EXTENSION 14.1 Options on Futures

14.4 Option Contract Features Maturity Dates Strike Prices Margin Requirements and Position Limits

Dividends and Stock Splits

14.5 Options Trading, Exercising, and the Expiration Process The Trading Process Exercising the Option The Expiration Process

14.6 Options Price Quotes 14.7 Regulation and Manipulation in Options Markets 14.8 Summary 14.9 Cases 14.10 Questions and Problems

EXCHANGE-TRADED OPTIONS

14.1

Introduction

Believe it or not, governments trade derivatives. Distinguished economist Lawrence Summers recounted an interesting experience when he was deputy secretary of the US Treasury (see Summers 1999, p. 3):

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Any doubt I might have had about the globalization of economic thinking was shattered when I met with Chinese Premier Zhu Rhongji in early 1997 in the same pavilion where Chairman Mao had received foreign visitors. After being offered a Diet Coke, I was asked a variety of searching questions about the possible use of put options [emphasis added] in defending a currency, and how they might best be structured. A communist leader dabbling in derivatives! A more recent example of derivative securities being used to support a government’s fiscal and monetary policy occurred in the United States during the 2007 credit crisis. To reduce the cost of the capital infusion by the US Treasury into financial institutions under the Troubled Asset Relief Program (TARP), warrants—a type of call option—were used. These warrants, issued to the US Treasury by the banks receiving the TARP funds, provided additional upside potential to the US Treasury. After the banks recovered, the US Treasury made significant profits on its warrant positions, reducing the cost of the TARP program. Exactly as hoped! This chapter continues our discussion of options and their markets started in Chapter 5. First, we review the history of options trading, which is divided into two phases around the watershed year of 1973. Next we describe some features of options contracts. We end the chapter with regulatory issues, including market manipulation. Although our discussions primarily revolve around equity options, many of the features discussed here are similar to other options encountered later in this book and elsewhere in life.

14.2

Exchange-Traded Options

Why trade options? Unlike futures, options have asymmetric payoffs, a special feature that allows users to carve out richer payoff structures than what’s otherwise obtainable. This makes options valuable tools for hedging, speculation, and risk management. Because the holder is rewarded when the underlying moves favorably but has no loss when it moves adversely, options are similar to insurance contracts and require a premium payment at their onset. But regular use of options can get expensive, and one must carefully weigh the costs and benefits before trading an option. We skip discussing the applications and uses of options because of its similarity with our earlier discussion of the applications and uses of derivatives (see section 1.3), swaps (see section 7.5), and forwards and futures (see section 8.2). Although options have traded over the counter (OTC) for hundreds of years, the market really took off when exchange-traded options were introduced on the

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Chicago Board Options Exchange (CBOE) in 1973. The reasons are twofold. First, an exchange creates a central marketplace: an orderly, efficient, and liquid market with low transaction costs where traders can enter or exit positions with ease. Second, a clearinghouse for the exchange guarantees contract performance and makes trades virtually free from credit risk. The resulting popularity of plain vanilla exchangetraded options paved the way for the development of the now gigantic OTC market for trading customized, complex options. The causation did not happen in reverse. Although OTC options have traded for centuries, their market remained small and inefficient, fraught with credit and legal risks, never catching on with the investing public until after exchange traded options became popular. Why did it happen this way? To find out, read on.

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14.3

A History of Options

“History has many cunning passages, contrived corridors,” wrote T. S. Eliot in the poem “Gerontion.” This describes the history of options. Option-like contracts arise naturally in economic life. One enters a call-like agreement when you pay a deposit in advance of buying a house with the provision that your failure to complete the purchase forfeits this amount; you enter into a put-like contract when you buy insurance by paying a premium. The term premium also denotes the price of an option, which reinforces the close link between options and insurance. Despite these similarities, the options and insurance markets went their separate ways. Insurance contracts have enjoyed popular support for centuries, but options’ acceptance has waxed and waned over the years. Options and futures traded side by side in Amsterdam during the Dutch golden age of the seventeenth century. They have traded for over two hundred years in small OTC markets in London, New York, and several cities of Continental Europe. Options traded as bilateral contracts that were loaded with credit, legal, and liquidity risks. They were viewed as tools for speculation that served no worthwhile economic purpose. In fact, the US Securities and Exchange Commission (SEC; created after the great stock market crash of 1929) repeatedly tried to outlaw options. The market ebbed and flowed depending on the participants. It expanded when big financiers wrote options because their reputation and deep pockets assuaged the buyers about credit risk. The opening of the CBOE in 1973, support from academic economists, and the Black–Scholes–Merton pricing model helped options reach the mainstream and remove their stigma of gambling. The information technology revolution of the 1990s ushered in the era of electronic exchanges and web-based trading. Insurance and options markets have converged again. Nowadays options are viewed as risk management tools that “complete the markets” by allowing traders to carve out different portfolio payoffs. They trade on many exchanges around the globe, and traders can transact through a single electronic platform.

A HISTORY OF OPTIONS

Despite this convergence, the history of options trading is sufficiently different from that of futures trading to merit further discussion. We divide the history into two phases: 1. Early trading of options. Option-like contracts have traded since antiquity, with options trading in organized OTC markets in Amsterdam, London, and New York City in past centuries. 2. The year 1973: the watershed year and after. The Nobel Prize–winning Black– Scholes–Merton model gave traders a vital tool for pricing and hedging options. The opening of the CBOE ended the old, inefficient OTC markets and brought numerous innovations. Post-1973 saw the development of a variety of contracts, the opening of new exchanges, the automation of trading, and the consolidation and linkages among exchanges. This story is akin to the history of futures trading discussed in Chapter 8. That’s hardly surprising because economic, political, regulatory, and technological changes similarly affected the options markets.

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Early Trading of Options Options and futures traded in Western Europe over four centuries ago. Before exploring that history, however, we mention that the first recorded option usage occurs in a story telling how Thales the Milesian (624–545 BC) purchased an option to buy the use of olive presses in the next harvest year, which he was sure would yield a bountiful harvest. His prediction was correct, and he profited handsomely from exercising his option.1 Let us move forward two millennia, and arrive in the Netherlands. Gelderblom and Jonker (2005) tell the story of how sophisticated financial markets developed in Western Europe, which, nonetheless, were vexed by problems that continue to afflict us to this day (see Table 14.1 for a timeline of options trading). Records from Amsterdam document how grain dealers in the city used both futures and options in the 1550s and the following decades. Fearing that speculators could use derivatives to manipulate staple food prices and cause social unrest, the authorities repeatedly banned forward trading. However, the market did not disappear; it simply went underground and was confined to the insiders. Subsequently, the market again became public, expanding to include forwards, futures, and call and put options on equities. The 1630s saw an extraordinary speculative frenzy over rare tulip bulbs, with futures and options helping hedgers and the speculating public. We discuss this episode later in this chapter. A small options market for sophisticated traders continued to operate in Continental Europe, London, and New York for over two centuries (see Malkiel and Quandt 1969). For example, during the 1690s, a well-organized options market existed in London, and it continued to exist despite the ban on options and futures promulgated under Barnard’s Act of 1733. In 1821, the London Stock Exchange Committee proposed a rule that would forbid members from dealing in options. Loud protest came from a large number of members; they even raised money to build a rival stock exchange building. The ban was never implemented, option trading continued 1

This story is from Jowett’s translation of Chapter 11 of Book 1 of Aristotle’s Politics (etext.library. adelaide.edu.au/a/aristotle/a8po/).

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TABLE 14.1: Early History of Options Trading Year

Development

600 bc onward

Thales of Miletus traded optionlike contracts

1550–1650

Birth of options and futures trading in Amsterdam

1630s

Futures and options traded on tulips in the Netherlands

1860–1900

Russell Sage extensively traded calls, puts, spreads, and straddles

1910

The Put and Call Brokers and Dealers Association was founded

(restricted but) uninterrupted, and Barnard’s Act itself was repealed in 1860. The European options market remained small because many prominent firms refused to participate. A New York City options market became active during the last four decades of the nineteenth century, when Russell Sage operated as a financier, securities trader, options writer, and broker on a vast scale. Russell Sage’s financing and trading activities relate to several important features of option markets:

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1. The importance of collateral. An article titled “Russell Sage” (New York Times, August 5, 1903), attributed Sage’s success to his being “an excellent judge of collateral.” It noted that “crises and panics have passed him unharmed, and when others were eager to borrow he was as eager to lend—if the collateral was satisfactory.” 2. The invention of straddles and spreads. Sarnoff (1965) gave Sage credit for inventing the spread and the straddle and suggested that his extensive dealing in calls and puts as well as these two creations led to the development of the “modern system of selling stock-option contracts.” These activities earned him the nickname “Old Straddle.” 3. Dividend adjustments. A dividend lowers the value of an asset; consequently, a dividend lowers the value of a call and raises the price of a put. A client once took the aged Sage to court in a dispute about dividend adjustments and an option’s value (see “Dividends and Puts,” New York Times, April 6, 1887). Today options exchanges have well-defined policies for these adjustments (see section 14.4 and Example 14.4 for further discussions). 4. Put–call parity. Sage devised conversions, where he created synthetic loans by using the put–call parity principle and effectively charged interest rates higher than what the usury laws would allow. The OTC option market in the nineteenth and the early twentieth centuries wasn’t too different from that in historic Amsterdam. Options contracts were written in printed forms where traders filled in the details and had them notarized. The dealers sold options by word of mouth or by advertising in the newspapers. The Put and Call Brokers and Dealers Association was established in 1910 to organize options trading in the United States.

A HISTORY OF OPTIONS

The government continued to view options trading with suspicion for good reasons. Particularly in the early twentieth century, speculators would often use options to manipulate stock prices. After the stock market crash of 1929, the US Congress sought to ban options trading, but did not do so. Options trading volume (in terms of shares), which was about half of 1 percent of the New York Stock Exchange (NYSE) stock trading volume in 1937, rose to around 1 percent in 1955 and hovered around that number during the 1960s; however, trading remained anemic, and the stigma of gambling prevented the options market from enjoying vigorous growth.

The Year 1973: The Watershed Year and After

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This landscape began to change in 1973 with the opening of the CBOE and the publication of the Black–Scholes–Merton options pricing model.

THE OPENING OF THE CHICAGO BOARD OPTIONS EXCHANGE The CBOE opened its doors on April 26, 1973. Just 911 call options contracts traded on sixteen stocks on the inaugural day. The trading floor was set up in the previous Chicago Board of Trade members’ smoking lounge. The daily trading volume at the end of the first month exceeded the average daily volume in the old, OTC options market. Despite its ups and downs, a remarkable expansion of options markets followed the opening of the CBOE—see Table 14.2 for some milestones in this extraordinary march. The mid-1970s saw the introduction of options trading on several US exchanges and in other nations. In 1975, the clearing facility of the CBOE was reorganized as the Options Clearing Corporation (OCC), which functions as the common clearinghouse for most exchange-traded listed options in the United States and has virtually eliminated credit risk. The same year, computerized price reporting was also introduced in the CBOE. During the three-year period 1977–80, the SEC put a moratorium on additional listings pending a review of the options industry’s growth. The stock market crash in October 1987 led to a temporary waning of interest in options. Still the innovations continued, and a variety of new contracts were introduced. The IT revolution of the 1990s led to several key developments: automation lowered transaction costs, many exchanges introduced electronic trading, greater links were established among exchanges, and a wave of mergers and consolidations transformed the markets. A VARIETY OF OPTIONS Most options varieties were introduced during the 1970s and 1980s. In 1977, put options started trading at the CBOE. The year 1982 saw the beginning of currency options trading on the Philadelphia Stock Exchange and Treasury bond futures options on the Chicago Board of Trade, which brought futures exchanges into the options business (see Extension 14.1). The following year, cash-settled options on broad-based stock indexes (including options on the Standard and Poor’s [S&P] 500 index) were launched at the CBOE. As with futures, cash settlement allowed the creation of new kinds of options with rich payoff structures.

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TABLE 14.2: Some Milestones in Options Markets, 1973 and After

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Year

Development

1973

First options exchange, the Chicago Board Options Exchange (CBOE), was founded, with call options trading on sixteen underlying stocks

1975

Call options began trading in the American Stock Exchange, the Philadelphia Stock Exchange, and the Montreal Stock Exchange in Canada; the Options Clearing Corporation (OCC) was established for exchange-traded options in the United States

1976

Calls began trading on the Pacific Exchange and the Australian Options Market

1977

Put options were introduced at the CBOE

1977–80

The Securities and Exchange Commission put a moratorium on the options market

1978

The European Options Exchange and London Traded Options Market were established

1982

Currency options started trading at the Philadelphia Stock Exchange

1982

Options on US Treasury bond futures started trading at the CBOT

1983

Options on broad-based stock indexes started trading at the CBOE

1987

Stock Market crash in October 1987 set back the options market, from which it took years to recover

1993

Introduction of the CBOE Volatility Index (VIX), a key measure of market expectations of near-term volatility conveyed by Standard and Poor’s index options prices

1998

Deutsche Borse AG and Swiss Exchange started Eurex, a joint platform for trading and clearing derivatives trade

2000

Euronext was founded through the merger and consolidation of several European exchanges

2000

First all-electronic exchange in the United States, the International Securities Exchange, was founded

By mid-1983, exchange-traded options were available for hedging and speculating on equity, commodity, currency, interest rates, and stock indexes. Many options were subsequently introduced in these categories. For example, the CBOE introduced long-lived equity options, LEAPS (Long-Term Equity AnticiPation Securities), with up to three years’ life in 1990, options on sector indexes in 1992, FLEX options (which allow traders to modify some key specifications) in 1993, and options on country indexes in 1994. On the basis of academic research by Professor Robert Whaley, the CBOE unveiled in 1993 the VIX (CBOE Volatility Index), which measures the market’s estimate of near-term volatility as implied by the S&P 500 stock index option prices. The VIX index was modified in 2003 from measuring an implied volatility to that of an expected volatility. Colloquially known as the “Fear Gauge,” VIX is a widely used barometer of investor sentiment and market volatility.

A HISTORY OF OPTIONS

295

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The CBOE introduced futures contracts on VIX in 2004 and option contracts on VIX in 2006—we discuss the VIX in chapter 20.

NEW EXCHANGES AND THE AUTOMATION OF TRADING Many existing exchanges embraced options trading, and new options exchanges were formed. In 1975, call options commenced trading at the American Stock Exchange, the Philadelphia Stock Exchange, and the Montreal Stock Exchange in Canada. In 1976, they were introduced on the Pacific Exchange in San Francisco and in the Australian Options Market, which became the first options market outside North America. In 1978, two of the world’s oldest exchanges introduced options trading in Europe: the European Options Exchange was founded by the members of the Amsterdam Stock Exchange, and the London Traded Options Market was established as a part of the London Stock Exchange. By the end of the decade, BOVESPA became the first exchange to introduce options trading in Brazil. During the 1980s, the exchanges worked on expanding the market and jockeyed for greater market share. The global options market suffered a severe setback during the stock market crash of October 1987, and it took over a decade for the volume to return to pre-crash levels. As the 1990s progressed, several discernible patterns emerged. Competition put relentless pressure to cut costs, and IT advances made that possible. Exchanges dismantled floor trading or began as completely electronic exchanges. With their commitment to floor trading, US exchanges kept falling behind. Wake-up calls came when Eurex and Euronext were founded in 1998 and 2000, respectively. Established in 2000 as an all-electronic options exchange, the International Securities Exchange (now part of Nasdaq Inc.) became the world’s largest equity options exchange. Stiff competition came from the OTC derivatives markets, which grew faster than the exchanges. Fed chairman Alan Greenspan (recall Chapter 10) used this fact to justify less regulation and a hands-off policy toward the OTC markets. The US exchanges regrouped and rethought their options (no pun intended!) and strategies. Seeing that electronic trading was the wave of the future for trading plain vanilla securities, they acquired or merged with organizations that focused on electronic trading or developed their own automated trading systems.

EXTENSION 14.1: Options on Futures Why trade options on futures? Recall the observation from Chapter 1 that trading gravitates to the most liquid markets with minimum transaction costs. As the stock market is very liquid with minimal transaction costs, it’s no surprise that we have bustling markets for equity options. But storage costs and convenience yields for many commodities in the spot market make it easier for traders to hedge and speculate on commodities in the more liquid futures markets. Consequently, it’s natural for options on futures to trade on such commodities. For options on spot (also called spot options), the spot commodity is obtained when the option is exercised. By contrast, exercise of a futures option gives the holder a position in the underlying futures. In the United

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States, active futures as well as futures options markets exist for diverse commodities such as cattle, cotton, corn, crude oil, gold, and silver. These options trade on the same exchange where the futures contracts trade. Simultaneous trading of futures and futures options helps the price discovery process. The example below shows how a futures option works. Notice that both spot options and futures options behave similarly in response to a movement in the underlying.

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EXT. 14.1 EX. 1: Call and Put Options on Futures ■

Today is January 1. Suppose the April gold futures price is $1,515 per ounce. Futures options with a contract size of one hundred ounces and for $1,490, $1,495, $1,500, $1,510, $1,520, $1,530, and several other strike prices (per ounce) trade in the New York Mercantile Exchange of the CME Group.

■

Consider a call option on a commodity futures. If you exercise this option, you will receive a long position in the underlying futures contract and a cash payment equal to the excess of the futures price over the strike price. For example, if you exercise an April 1,500 call, you end up long an April futures and receive a payment of (1,515 − 1,500) = $15 per ounce, $1,500 in all. If the payment is negative, do not exercise—enter the futures at zero cost in the market instead.

■

The exercise of a put option on a commodity futures puts the holder into a short position in the underlying futures and brings in a cash amount equal to the strike price minus the futures settlement price. For example, if you exercise an April 1,520 put, you end up being short April futures and receive a payment of (1,520 − 1,515) = $5 per ounce.

■

Obviously, the long has to pay premiums for these options. A modified version of the Black–Scholes–Merton model can be used to price futures options. More advanced models like the Heath–Jarrow–Morton model (developed in 1987, published in 1992) can also be used for this purpose (see Amin and Jarrow 1991).

Notice that if you replace the futures price F(T) with the stock price S(T), then futures options payoffs are similar to equity options payoffs. For example, consider a call futures option and a regular equity call option on a stock with the same strike price K. If you exercise the equity call, your payoff is [S(T) − K]. If you exercise the call futures option, you receive [F (T) − K] and a long futures position on the stock. As the futures contract has a zero value, it can be liquidated at no cost. Exercise of a put futures option leads to a short position in the underlying futures plus the strike minus the futures price, [K − F (T)], and a similar argument holds.

14.4

Option Contract Features

Let us examine the contract specifications for CBOE equity options.2 The CBOE regular trading session for options is from 8:30 am to 3:00 pm Central Time (Chicago time). CBOE equity options are American, and they can be exercised any time from purchase until the option’s last trading day, which is the third Friday of the expiration month. If the third Friday is a legal holiday, then the option expires the day before. Each option is for one hundred shares of the underlying stock. If exercised, the stock share is delivered three business days later. 2

See www.cboe.com.

OPTION CONTRACT FEATURES

The minimal tick size is $0.05 for options trading below $3 and $0.10 for options trading above. However, options series are quoted in pennies ($0.01) for options prices below $3 and in nickels ($0.05) for higher options prices.

Maturity Dates For a particular stock, options belong to an assigned quarterly cycle, either the January, February, or March cycle. With respect to a particular cycle, options maturities are listed for “two near-term months plus two additional months from the January, February or March quarterly cycles (www.cboe.com).” An example explains this procedure.

EXAMPLE 14.1: Determining the Expiration Months ■

The January cycle has the months January, April, July, and October. The February cycle has February, May, and so on. Each stock approved for options trading belongs to one of the cycles. For example, IBM has a January cycle, Hewlett Packard has a February cycle, and Walmart Stores has a March cycle.

■

Consider the expiration months for traded options on IBM starting January 1. - On January 1, IBM has options expiring in January, February (the two near-term months), April, and July (the next two months from the quarterly cycle). - After the third Friday of January, one can trade IBM options maturing in February, March, April, and July. - After the third Friday of February, one can trade options maturing in March, April, July, and October.

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

Different kinds of options have different expiration dates. For example, LEAPS are issued with threeyear maturities and expire on the third Friday of the expiration year. Quarterly options (or the “quarterlies”), which can be written on some indexes and exchange-traded funds, expire on the last business day of each calendar quarter. Weeklys, which can be traded on many stocks, ETFs, and indexes, typically expire on Friday of each week.

Strike Prices When an option with a particular maturity starts trading, strike prices for in-, at-, and out-of-the-money are listed. If the stock price is over $200, the strikes are issued $10 apart. Between $200 and $25, the strikes are issued $5 apart, and below $25, only $2.50 apart. The CBOE has also introduced $1 strike price intervals and $2.50 strike price intervals, under certain terms and conditions, for many actively traded stocks. Strike prices are adjusted for stock dividends and stock splits but not for cash dividends. If the stock price moves significantly before the option expires, then new strikes are listed to maintain the balance with respect to in-, at-, and out-of-themoney. The next example illustrates this process.

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EXAMPLE 14.2: Determining the Strike Prices ■

Suppose the stock opens at $21 on Monday following the third Friday of the month, when new long-maturity options are listed. This is the first trading day after the shortest maturity option expires. Newly listed options (both calls and puts) have strike prices of $20 and $22.50, immediately above and below the current stock price. Additional in- and out-of-the-money strikes are also listed at this time.

■

Next, suppose that only the $20 and $22.50 options are listed, and the stock rallies. When it reaches $22.51, options with a strike price of $25 are introduced. If it goes beyond $25, options with the next higher strike of $30 start trading, and so forth. Of course, one can still trade options with strike prices $20 and $22.50, which were listed earlier, until these contracts expire.

■

Liquidity is usually the greatest for options with strike prices near the current stock price. Trading volume and the number of contracts outstanding tend to decline as the stock goes deeper in- or out-of-the-money.

Margin Requirements and Position Limits

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

The margin (borrowing) restrictions for long positions in options can be summarized as follows: (1) short-term options (nine months or less) must post the full purchase price with zero borrowing and (2) long-term options (more than nine months) can borrow up to 25 percent of the purchase price. Options sellers are usually required to keep the full purchase price plus some extra cash as a cushion against risk. An example illustrates these restrictions.

EXAMPLE 14.3: Margin Account Adjustments ■

Long call or long put (maturing in less than nine months). For example, from Figure 14.1, if Ms. Long is buying five IBM March 2009 (IBMCR) calls worth $2.30 per share, she will need to keep (Ask price × Number of contracts × Contract size) = 2.30 × 5 × 100 = $1,150 in her margin account.

■

Long call or long put (maturing in over nine months). The initial (maintenance) margin requirement is 75 percent of the cost (market value). For example, if Ms. Long wants to buy ten IBM January 2010 (WIBAR) contracts worth $12.50, she will have to keep at least 75 percent of 12.50 × 10 × 100 = 0.75 × 12,500 = $9,375 of her own money as margin and borrow up to $3,125 from her broker.

■

Options writer. Suppose Mr. Short is selling ten IBM October 95 (IBMVS) put options (see Figure 14.1). Then margin would be computed as follows: - 100 percent of contract proceeds (13.30 × 10 × 100) $13,300 - Plus 20 percent of aggregate contract value (0.2 × 90.42 × 1,000) 18,084 - TOTAL amount (13,300 + 18,084) $31,384 Also compute - Put option proceeds (13.30 × 10 × 100)

$13,300

OPTION CONTRACT FEATURES

- Plus 10 percent of aggregate exercise price (0.1 × 95 × 1,000) - TOTAL amount (13,300 + 9,500)

9,500 $22,800

The margin Short has to keep is $31,384, which is the larger of the two.

If the options holder-writer has positions in other options or the underlying, the CBOE adjusts these margin restrictions, taking into account the portfolio’s overall risk. For example, the margin required for a long stock plus a long put position is significantly lower than a long put position alone. The adjusted margin takes into account the reduced risk from the long stock hedging the long put position. Options position limits are imposed to reduce the potential for market manipulation. For actively traded stocks with large capitalizations (e. g., IBM or Microsoft), the position limit is 250,000 contracts. Stocks with smaller capitalizations have smaller position limits. Exemptions may be granted for qualified hedging strategies.

Dividends and Stock Splits

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Options are not adjusted for cash dividends or stock distributions that do not exceed 10 percent of the stock’s value. When a stock splits or pays large stock dividends, the option’s strike price is adjusted on the ex-dividend day or the ex-split day. The adjustment depends on whether the split or stock dividend is an integer multiple such as 3 for 1 or a fractional such as 3 for 2. The OCC has developed these rules and procedures in an attempt to guarantee that options traders do not suffer losses from these events.

EXAMPLE 14.4: Options Adjustment for Stock Splits and Stock Dividends IBM’s stock price is $90.42 (see Figure 14.1). Consider the April 90 call contract.

An Integer Multiple split ■

Suppose IBM has a 3 for 1 split. An investor who held one share would now have three shares worth $30.14 per share. On the ex-distribution day, the number of options will increase by a factor of 3, and the strike price will be reduced by a factor of 3.

■

The aggregate exercise price before the split was 100 × $90 = $9,000. After the split, it equals 3 × 100 × $30 = $9,000. They are the same—the adjustment makes the option neutral with respect to a stock split.

A Fractional split ■

Suppose IBM has a 3 for 2 split. An investor holding two shares will now own three shares. To prevent arbitrage, the new share’s price will be 90.42/1.5 = $60.28, where 3/2 = 1.5 is the split ratio.

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■

In this case, instead of adjusting the number of contracts, the number of shares in each contract is multiplied by the split ratio: 100 × 1.5 = 150. The new strike price is obtained by dividing the old strike price by the split ratio. Thus the new exercise price after the split is 90/1.5 = $60.

■

Notice that the aggregate exercise price before the split was $9,000. After the split, the aggregate exercise price is 150 × $60 = $9,000. Again, this adjustment makes the option neutral with respect to the stock split.

Stock Dividends ■

Large stock dividends (greater than 10 percent of the stock’s price) are called stock splits and are treated accordingly. For example, a 3:1 split is really a 200 percent stock dividend, and a 3:2 split is a 50 percent stock dividend.

Reverse Split ■

If IBM has a 1 for 2 reverse split, the stock price would become $180.84. The contract is adjusted so that each original option is now an IBM April 180 call covering fifty shares.

14.5

Options Trading, Exercising, and the Expiration Process

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

The Trading Process Options trading involves the three-step process of execution, clearing, and settlement inherent to exchange-traded securities. As with futures, only a clearing member can clear trades. A clearing member must be a member of an options exchange, satisfy OCC’s minimum capital, and meet other requirements. He must maintain efficient operations on the trading floor and in the back office. Clearing members pay both the exchange and the OCC a fee for each traded contract. They recoup these expenses from the fees charged to their customers.

Exercising the Option The process begins when the holder conveys her intention to exercise the option to her broker. To ensure exercise on a particular day, the notification must be given before the broker’s daily cutoff time. Next, the broker submits an exercise notice to the OCC. The OCC sends the notice to one or more clearing members holding short positions on the same contract. The clearing member, in turn, assigns it (either randomly or on a first-in, first-out basis) to one or more customers who hold short positions in the contract. The clearing member representing the short is obligated to sell (in the case of a call) or buy (in the case of a put) the underlying shares at the specified strike price. The OCC then arranges for the delivery of the shares of stock and the exchange of strike price funds.

OPTIONS PRICE QUOTES

The Expiration Process The expiration date is the last day an option trades. For listed stock options, this is the third Friday of the expiration month. Brokerage firms must submit exercise notices to the OCC. The OCC has developed a procedure known as exercise by exception (also called ex-by-ex) to expedite the handling of the exercise of expiring options. Every contract in-the-money will be automatically exercised unless the clearing member advises OCC otherwise. Currently equity and index options that are in-the-money by at least 1 cent are automatically exercised. Nonetheless, most brokers still require their customers to notify their intention to exercise even if an option is in-the-money. When an equity option is exercised or assigned and it is not covered by stock holdings in the account, sufficient purchasing power must be in the account to make or take delivery of the shares. Otherwise, the broker may close your expiring options positions, even without notification, on the option’s last trading day. If the expiring position is not closed and there is insufficient purchasing power, the broker may submit “do not exercise” instructions to the clearing member without notification.

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

14.6

Options Price Quotes

Wire news services, websites, and many major daily newspapers carry options price quotes. Because each stock has calls and puts that differ in terms of exercise price and expiration months, an exchange typically has tens of thousands of options trading. All options contracts of the same type written on the same underlying stock belong to the same option class. For example, different IBM calls belong to one option class, whereas different IBM puts belong to another. Within a particular class, all options with the same strike price and expiration date belong to an option series; for example, IBM March 100 calls belong to one option series, IBM April 90 calls belong to another option series, and IBM April 100 puts belong to yet another option series. Example 14.5 illustrates IBM’s options price quotes.

EXAMPLE 14.5: IBM Options Price Quotes ■

IBM’s stock price is $90.42, which is up $1.80 (2.03 percent) from the previous day’s close. Figure 14.1 gives the IBM options “chain table” data from March 12, 2009, compiled from the CBOE website (the figure has been condensed and all quotes are twenty minutes delayed). Calls are written on the left side and puts are written on the right side of the figure, with the strike price in the middle. The first set of numbers reports call and put prices for March 2009, with the shaded region reporting options that are in-the-money.

■

Consider the tenth entry under “Calls,” for which $90 is the strike price. The row contains the following data, moving from left to right: - This option is also identified by IBMCR—C stands for the third month, March, and R stands for the strike $90.

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-

The price $2.27 is recorded under “Last Trade,” which stands for an option’s last reported trade price. The $0.56 denotes the price change from the previous settlement price. The bid price $2.25 and the ask price $2.30 are reported per option on one IBM share. The 6,516 under “Volume” is the number of contracts that have traded so far during the trading day. “Interest” 14,198 is the open interest, which is the total number of outstanding contracts.

FIGURE 14.1: IBM Options Quote from CBOE Website (IBM) International Bus Mach Corp Com March 2009 Calls

Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Symbol

Puts

Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade

IBMCX 43.40

2.40

45.20 45.60

n.a.

n.a.

45.00 IBMOX

0.10

0.00

0.05 0.05

35

123

IBMCU 36.00

−6.60

40.30 40.60

n.a.

n.a.

50.00 IBMOU 0.05

0.00

0.05 0.05

148

2,350

IBMCV 33.40

4.40

35.20 35.60

n.a.

5

55.00 IBMOV 0.05

0.00

0.05 0.05

7

325

IBMCL 25.20

1.20

30.20 30.60

7

92

60.00 IBMOL

0.05

0.00

0.05 0.05

7

827

IBMCM 20.90

−3.00

25.30 25.60

4

117

65.00 IBMOM 0.05

0.00

0.05 0.05

34

1,309

IBMCN 19.60

0.60

20.30 20.60

14

218

70.00 IBMON 0.05

0.00

0.05 0.10

20

3,153

IBMCO 12.80

−1.70

15.40 15.60

30

578

75.00 IBMOO 0.09

−0.01

0.05 0.10

307

7,409

IBMCP 10.45

1.35

10.50 10.70

116

2,874

80.00 IBMOP 0.20

−0.05

0.15 0.20

828

10,548

IBMCQ

5.90

1.10

5.80

6.00 1,200

10,054

85.00 IBMOQ 0.50

−0.40

0.50 0.55 5,373

14,901

IBMCR

2.27

0.56

2.25

2.30 6,516

14,198

90.00 IBMOR 1.85

−1.05

1.80 1.90 2,741

12,079

IBMCS

0.50

0.15

0.45

0.50 3,078

14,336

95.00 IBMOS

5.00

−1.00

5.00 5.10

426

4,213

IBMCT

0.05

−0.01

0.05

0.10

148

10,760 100.00 IBMOT 10.10

−1.49

9.50 9.70

62

1,913

IBMCA

0.05

0.00

0.05

0.15

2

4,150 105.00 IBMOA 15.30

−4.10 14.50 14.80

6

75

IBMCB

0.03

−0.02

0.05

0.05

16

3,473 110.00 IBMOB 18.30

−7.00

19.40 19.80

26

10

IBMCC

0.05

0.03

0.05

0.05

n.a.

410

115.00 IBMOC n.a.

n.a.

24.30 24.70

n.a.

n.a.

IBMCD

0.03

−0.04

n.a.

0.05

n.a.

81

120.00 IBMOD n.a.

n.a.

29.50 29.80

n.a.

n.a.

IBMCE

n.a.

n.a.

n.a.

0.05

n.a.

n.a.

125.00 IBMOE 34.00

0.00 34.40 34.70

1

n.a.

OPTIONS PRICE QUOTES

Apr 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade 11.8

0.90

11.60 11.80

458

IBMDQ 7.80 IBMDR 4.60

0.99

7.70 7.90

2,381

9,649

85.00 IBMPQ 2.40

0.73

4.50 4.70

1,672

11,644

90.00 IBMPR 4.20

2.40

0.60

2.30 2.40

3,405

21,552

95.00 IBMPS 6.90

IBMDT 1.05

0.30

0.95 1.05

992

IBMDP

IBMDS

3,221

80.00 IBMPP 1.30

15,196 100.00 IBMPT 12.01

−0.25 1.25 1.35

−0.23 2.40 2.45

−0.75 4.10 4.30 −1.20 6.80 7.00

−3.19 10.5010.70

1,527

12,260

1,060

13,098

2,145

10,644

254

14,501

1

2,553

Jul 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade IBMGP 14.23

0.83

14.50 14.70

IBMGQ 11.40

1.00

IBMGR 8.50

0.80

6.10

IBMGT 4.22

IBMGS

88

1,605

80.00 IBMSP 4.70

−0.26 4.60 4.70

58

3,457

11.20 11.40

317

8.40 8.50

600

1,883

85.00 IBMSQ 6.30

−0.60 6.30 6.40

264

4,300

5,110

90.00 IBMSR 8.67

−0.53 8.40 8.60

93

5,381

0.80

6.00 6.10

168

0.62

4.10 4.20

274

8,500

95.00 IBMSS 11.00

−0.70 11.0011.20

27

1,283

6,417

100.00 IBMST 15.31

−0.19 14.10 14.30

n.a.

848

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Oct 2009 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade IBMJP 16.23

2.33

16.30 16.60

20

397

IBMJQ 13.40

2.70

IBMJR 10.59

2.29

80.00 IBMVP 7.00

−1.86 6.60 6.80

2

162

13.20 13.50

3

202

85.00 IBMVQ 9.30

−1.70 8.50 8.70

12

345

10.50 10.70

48

119

90.00 IBMVR 11.32

−0.18 10.70 10.90

n.a.

387

IBMJS

7.40

1.10

8.10 8.30

44

519

95.00 IBMVS 14.40

−0.60 13.3013.60

44

257

IBMJT

6.00

0.40

6.10 6.30

13

273

100.00 IBMVT 17.90

−2.30 16.3016.60

43

98

Jan 2010 Calls

Puts

Symbol Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade WIBAP 17.70

2.30

17.60 18.30

15

1,744

80.00 WIBMP 8.60

−1.80 8.40 8.70

n.a.

5.576

WIBAQ 14.10

2.30

14.60 15.20

1

1,151

85.00 WIBMQ 11.70

−0.90 10.40 10.70

8

2,616

WIBAR 11.63

1.23

12.20 12.50

n.a.

7,696

90.00 WIBMR 13.20

−1.60 12.7013.00

2

6,936

WIBAS

9.50

1.70

9.90 10.10

3

3,383

95.00 WIBMS 15.70

−0.90 15.3015.70

47

2,008

WIBAT

7.74

0.27

7.80 8.10

6

10,373 100.00 WIBMT 19.60

−3.70 18.3018.60

22

5,714 (continued)

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CHAPTER 14: OPTIONS MARKETS AND TRADING

Jan 2011 Calls Symbol

Last Change Bid Ask Volume Interest Strike Symbol Last Change Bid Ask Volume Interest Trade Price Trade

VIBAU

n.a.

n.a.

69.30 70.80

n.a.

n.a.

20.00 VIBMU 0.45

0.00

0.30 0.45

50

358

VIBAX

n.a.

n.a.

67.00 68.20

n.a.

n.a.

22.50 VIBMX 0.55

−0.25

0.45 0.60

100

204

VIBAE

n.a.

n.a.

64.50 65.80

n.a.

n.a.

25.00 VIBME 0.80

−0.12

0.65 0.75

2

105

VIBAV

n.a.

n.a.

59.30 60.80

n.a.

n.a.

30.00 VIBMV 1.20

−0.25

1.05 1.25

n.a.

883

VIBAG 49.50

1.00

54.70 55.90

n.a.

25

35.00 VIBMG 2.10

−0.08

1.65 1.90

20

163

VIBAW 47.70

1.90

50.10 51.30

1

41

40.00 VIBMW 2.95

0.05

2.40 2.60

1

236

−6.20 45.80 47.00

2

7

3.40

−0.10

3.20 3.60

n.a.

147

VIBAZ 39.20

−6.60

41.60 43.00

10

207

50.00 VIBMZ 4.70

−0.80

4.20 4.60

10

407

VIBAK 35.40

0.00

37.80 39.00

2

2

55.00 VIBMK 6.30

0.24

5.40 5.90

10

135

−0.10 34.30 35.40

3

92

60.00 VIBML 7.00

−0.80

6.80 7.20

1

323

VIBAM 28.20

−5.30

2

21

65.00 VIBMM 9.60

−0.70

8.30 8.70

44

293

VIBAN 24.80

−0.70 28.00 28.80

n.a.

254

70.00 VIBMN 11.60

0.00

10.00 10.60

25

331

VIBAP 20.20

−2.84 22.00 22.90

1.50

14.10 14.70

40

439

−0.96 16.30 16.90

117

748

18.80 19.40

n.a.

1,107

−1.90 21.50 22.20

11

636

VIBAI

VIBAL

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Puts

42.90

31.50

31.40 31.90

45.00 VIBMI

6

293

80.00 VIBMP 16.70

VIBAQ 18.50

0.96

19.90 20.30

50

479

85.00 VIBMQ 18.80

VIBAR 15.80

0.40

17.20 17.90

10

1,079

90.00 VIBMR 22.40

14.30

1.20

14.90 15.70

45

521

95.00 VIBMS 22.60

VIBAT 13.00

0.40

13.00 13.70

1

1,424

100.00 VIBMT 27.00

0.10

24.40 25.20

n.a.

676

VIBAB

9.80

0.30

9.50 10.10

4

812

110.00 VIBMB 35.50

0.10

30.70 31.60

24

512

VIBAD

6.60

−0.20

6.90 7.50

34

638

120.00 VIBMD 42.90

2.10

37.70 38.70

11

196

VIBAS

1.40

VIBAF

4.80

0.40

4.90 5.30

16

407

130.00 VIBMF 47.00

−4.00 45.40 46.20

3

193

VIBAH

3.60

0.60

3.20 3.70

1

599

140.00 VIBMH 54.80

−0.10 53.60 54.70

47

257

VIBAJ

2.25

0.05

2.25 2.50

67

672

150.00 VIBMJ 66.50

−2.00 62.30 62.90

42

227

Source: http://delayedquotes.cboe.com/options/options_chain.html?ASSET_CLASS=STO&ID_OSI=85502&ID_NOTATION=1551887

REGULATION AND MANIPULATION IN OPTIONS MARKETS

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14.7

Regulation and Manipulation in Options Markets

In the movie Mission: Impossible II (2000), a fictional character, Sean Ambrose, plans to release a virus and then profit by selling the antidote. “I want stock!” Sean thunders. “Stock options, to be precise.” And in the James Bond movie Casino Royale (2006), a fictional banker, Le Chiffre, provides a guerrilla group a safe haven for its funds. But Le Chiffre’s investments actually involve considerable risk. He buys puts on a successful company and then engineers a terrorist attack to sink the stock values. Stock options and market manipulation have entered pop culture! Options have long been associated with market manipulation. For example, during the 1920s, options gained notoriety when they were used in conjunction with stock price manipulation schemes. One abusive practice was for speculators with large holdings of the underlying stock to grant call options to stockbrokers (see Overby 2007). In return, brokers would recommend their clients purchase the stock. Another market manipulation scheme is buying out-of-the-money options and simultaneously buying the underlying shares to artificially increase the stock price so that the options end up in-the-money. Traders on the other side, who were getting ready to go out and celebrate happy hour on a late Friday afternoon, may suddenly find that their short out-of-the-money call options, which were seemingly worthless, suddenly end up in-the-money! We introduced the notion of a price bubble in Chapter 11, which happens when the stock price substantially deviates from its intrinsic or fundamental value. Bubbles typically involve a dramatic price rise followed by a drastic decline. Bubbles are associated with the dotcom boom of the 1990s and the recent real estate rise and collapse, and options are often connected with such episodes. A classic case was the tulip bulb bubble in seventeenth-century Holland. Dutch tulips are famous to this day, but their early history was fraught with speculation, greed, and fraud (see Malkiel 2003). The tulip bulb bubble story begins in 1593, when a botany professor brought tulip bulbs from Turkey to Holland, hoping to vend them at high prices. This market flourished, and tulips traded actively in the Netherlands in the years that followed. Some time later, many bulbs caught a benign virus that patterned their petals with bright stripes of contrasting colors. The public adored these “bizarres” and bid their prices up. Tulip bulb prices increased to a peak during the mid-1630s, when people from all walks of life, from noblemen to chimney sweeps, joined the fray. Some even bartered or pawned personal belongings to buy tulip bulbs with the hope of selling them at ever-increasing prices. Derivatives sprang up to feed the frenzy, and speculators used these derivatives (including options) for leverage—but they also helped the hedgers. Tulip growers and retailers bought calls and futures for protection against price increases. Such hedging strategies are practiced by many businesses even today. After months of outrageous prices and tumultuous trading, the market suddenly crashed in February 1637. Bad regulation and poor quality control have been blamed as the reasons for the crash. It’s no surprise that the regulators have historically been hostile to options. MacKenzie and Millo (2003) note that the proposal for the establishment of the

305

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CBOE was met with “instinctual hostility, based in part upon corporate memory of the role options had played in the malpractices of the 1920s.” CBOE’s first president, Sullivan, was told by one leading SEC official that he had “never seen a [market] manipulation” in which options were not involved. A former SEC chairman even compared options to “marijuana and thalidomide.” In a twist of fate, today’s regulators support options exchanges and consider them the lesser evil compared to the unregulated OTC derivatives markets.

14.8

Summary

1. Options trade because they provide payoffs different from forwards, futures, and the underlying stock. 2. Historically, options have been looked at with suspicion because of a cumbersome transaction process, small trading volume, high fees, counterparty risk, and the bad repute of market participants. The result was that investors surreptitiously traded a small number of contracts or shunned them altogether. The government’s response ranged from a grudging acceptance to an outright ban.

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3. Options markets got a boost in 1973 with the publication of the Black– Scholes– Merton model and the opening of the CBOE. Options trading has seen tremendous growth in the years that followed. This includes trading in options on interest rates, sector funds, exchange-traded funds, and indexes. 4. Equity options traded on the CBOE have the following features: a. Nature: American physical delivery options b. Contract size: one hundred shares c. Last exercise date: the third Friday of the expiration month d. Quotes: usually quoted in pennies ($0.01) for options prices below $3 and in nickels ($0.05) for higher options prices e. Strike prices intervals: $2.50 when the strike price is between $5 and $25, $5 between $25 and $200, and $10 for points over $200; strikes are adjusted for splits and recapitalizations but not for cash dividends (the CBOE has also introduced $1 strike price intervals and $2.50 strike price intervals, under certain terms and conditions, for many actively traded stocks) f. Strike prices: in-, at-, and out-of-the-money strike prices are initially listed; new series are generally added when the underlying trades through the highest or lowest strike price available g. Expiration months: two near-term months plus two additional months from the January, February, or March quarterly cycles h. Margin: short-term options buyers post the full price; long-term options buyers (maturing in more than nine months) can borrow up to 25 percent of the purchase price; options sellers are required to keep the full purchase price plus extra funds as cushion against risk

QUESTIONS AND PROBLEMS

14.9

Cases

Chicago Board Options Exchange (CBOE) (Harvard Business School Case

205073-PDF-ENG). The case discusses institutional details behind how options trade at the CBOE. International Securities Exchange: New Ground in Options Markets (Harvard

Business School Case 203063-PDF-ENG). This case examines the equity options market, the major parties involved, and the options trading process. Milk and Money (Kellogg School of Management Case, Case KEL343-PDF-ENG,

Harvard Business Publishing). This case considers how a family dairy firm can use regression analysis to choose the best hedges for its dairy products.

14.10 Questions and Problems 14.1. Briefly describe options trading that took place in 1500–1700 Amsterdam and

in nineteenth-century London and New York City. 14.2. a. Who was Russell Sage? b. Why was Sage called “The Old Straddle”? 14.3. Why did options trading fall into disrepute in the United States during the

early decades of the twentieth century? 14.4. During the nineteenth century, futures trading in the United States steadily

gained acceptance while options trading was associated with suspicion. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

a. Can you explain why this happened? b. How and when did this view of options trading get changed? 14.5. Briefly describe the events and developments that led to the founding of the

CBOE. 14.6. Describe two developments that took place in 1973 that made it a watershed

year in the history of options. 14.7. What is an option on a futures? Explain the workings of this derivative

contract. The next two questions are based on the following data from the CME Group’s website (prices as of 6:59:36 pm Central Standard Time on August 26, 2011). Spot price of gold is $1,830 per ounce. Contract size is one hundred ounces. Prices are per ounce. 14.8. a. Identify which calls are in-the-money, at-the-money, and out-of-the-

money. b. If you exercise a call with a strike price of $1,820, what is your payoff, and

what are your holdings of the futures contracts? c. For this call option on gold futures, what is the intrinsic value, and what

is the time value?

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Strike

Call

Put

1,820

85.20

107.80

1,825

83.20

110.90

1,830

81.30

113.90

1,835

79.40

217.10

1,840

77.50

120.20

14.9. a. There is an obvious mistake in the put price data—correct that first. b. Identify which puts are in-the-money, at-the-money, and out-of-the-

money. c. If you exercise a put with a strike price of 1,835, what is your payoff, and

what are your holdings of the futures contract? d. For this put option on gold futures, what is the intrinsic value, and what

is the time value? 14.10. Suppose Your Beloved Machines Inc. has a February cycle for options

trading.State the months for which regular equity options on YBM (which expire on the third Friday of the month) trade on the following dates: a. January 1

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b. January 27 c. March 1 14.11. State the dollar amount of margin you are required to keep with a broker

when trading one contract (on one hundred shares) of the following options on Your Beloved Machines Inc. YBM’s current stock price is $103. a. A long put worth $6 for an option maturing in six months b. A long call worth $8 for an option maturing in ten months 14.12. A long is generally associated with buying and a short with selling. Is it

counterintuitive that the put holder gets the right to sell? Explain your answer. 14.13. Your Beloved Machine’s current stock price is $90. YBM December 100 calls

trade for $6. a. Adjust the options prices and terms of the contract for a 4:1 split. b. Adjust the options prices and terms of the contract for a 3:2 split. 14.14. Are options on the CBOE adjusted for cash dividends or stock dividends or

both? 14.15. Use Figure 14.1. a. For the IBM April 2009 calls, is the call value increasing or decreasing in

the strike price?

QUESTIONS AND PROBLEMS

b. For the IBM April 2009 puts, is the put value increasing or decreasing in

the strike price? c. For the IBM March 2009 calls, which strikes are the most actively traded? d. For the IBM March 2009 puts, which strikes are the most actively traded? 14.16. What is exercise by exception with respect to CBOE options? 14.17. What is an options class? What is an options series? Give examples to illustrate

your answer. 14.18. What is open interest for a traded options contract? What does this tell you

about the trading interest in a particular options contract?

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14.19. Give two examples of market manipulation in the options market.

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15 Option Trading Strategies 15.1 Introduction 15.2 Traders in Options Markets 15.3 Profit Diagrams Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

15.4 Options Strategies

15.6 Spread Strategies EXTENSION 15.1 Collars, Ratio Spreads, Butterflies, and Condors EXTENSION 15.2 Reinsurance Contracts and Call Spreads

Common Options Price Data

15.7 Combination Strategies

Naked Trades: The Basic Building Blocks

15.8 Summary

15.5 Hedged Strategies

15.9 Cases 15.10 Questions and Problems

TRADERS IN OPTIONS MARKETS

15.1

Introduction

While teaching options and futures for the first time, one of us (Chatterjea) was sharing some frustrations with a colleague, Mike Goldstein, about the difficulties in motivating students. Mike advised, “That’s not the way to teach derivatives. You should cover the interesting stuff, and show how to implement option trading strategies. Suppose that a takeover is going on, and explain how to set up a straddle to bet on the outcome.” He had hit the nail on the head! Focusing on equations like C(K1 ) – C(K2 ) ≤ K1 – K2 for K1 ≥ K2 is not the best way to attract interest. Moreover, this gives a false sense of mathematical sophistication while achieving little substance. It’s far more exciting to take options where the action is and discuss their role for speculating on takeover battles, placing leveraged bets on the markets, and protecting portfolios from market crashes. This is the approach we take here. This chapter studies some popular options trading strategies. We present profit diagrams for the basic options trades, hedged and spread strategies, including “butterflies” and “condors,” followed by strategies involving combinations of options. Along the way, we discuss how options relate to insurance contracts and present a bird’s-eye view of taxation and reporting requirements. After this chapter, you should be able to understand option related stories in The Wall Street Journal. The strategies we discuss are relevant to all players in the options market: the individual trader, the financial institution, the nonfinancial corporation, and even the government.

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15.2

Traders in Options Markets

Bill Lipschutz, who had taken one of the world’s first options courses from one of us (Jarrow) at Cornell University, joined Salomon Brothers Inc. in 1981, a prestigious Wall Street investment banking firm at the time.1 Though Salomon prided itself as one of the world’s most powerful trading companies, Lipschutz was disappointed to find options trading to be nonquantitative. He found that no one seemed to know the Black–Scholes–Merton model. Trading strategies were based on ad hoc criteria, for example, the head of the proprietary options trading desk said, “‘I went to buy a car this weekend and the Chevrolet showroom was packed. Let’s buy GM calls.’ That type of stuff.” Then, one day, a trader pulled him aside and gave away the great secret: “Look, I don’t know what Sidney is teaching you, but let me tell you everything you need to know about options. You like ’em, buy calls. You don’t like ’em, buy puts.” In the early 1980s, many Wall Street firms placed “leveraged outright bets” and prayed to the “Bitch-Goddess Success.”2 As the decade unfolded, however, most investment banks started assembling a team of derivative experts and running 1

This story is taken from the interview “Bill Lipschutz: The Sultan of Currencies” in Jack Schwager’s (1992) book The New Market Wizards: Conversations with America’s Top Traders. Lipschutz developed an early expertise in trading currency options, which began trading on the Philadelphia Stock Exchange in 1982. During his eight-year stint at Salomon, he routinely traded billions of dollars’ worth of currencies and became the “largest and most successful currency trader” in the firm. 2

Professor William James of Harvard University used the expression “Bitch-Goddess Success” in a letter to the British writer H. G. Wells.

311

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CHAPTER 15: OPTION TRADING STRATEGIES

proprietary trading desks to make profits. Goldman Sachs and Company, a key player in this transformation, lured Professor Fischer Black from the Massachusetts Institute of Technology and made him a partner. Black helped usher in the migration of many finance professors and numerous talented PhDs in the natural and engineering sciences into Wall Street firms. The most mathematical of these recruits came to be variously viewed as “quants” and “rocket scientists” whose job was to apply their math mastery to make profits. Salomon did not lag behind. It pioneered the mortgagebacked securities market during the 1970s, arranged the world’s first swap contract in 1981 (see Chapter 22), and built an active derivatives shop. Its consultants included Nobel laureate Robert Engle, who was hired to develop volatility forecasts and devise trading strategies based on cutting-edge econometrics. Much of our discussion of options strategies in this chapter will be from the perspective of an individual investor. Financial institutions also trade options to bet on the markets and capture arbitrage profits. Hedge funds and proprietary trading desks at investment banks also implement the strategies we describe. Warren Buffett noted in his 2008 chairman’s letter in the annual report of Berkshire Hathaway Inc. that his company had entered into 251 derivatives contracts, including writing fifteen-and twenty-year maturity European put options on four major indexes (the Standard and Poor’s [S&P] 500, London’s FTSE 100, Europe’s Euro Stoxx 50, and Tokyo’s Nikkei 225). He believed that each of these contracts was “mispriced at inception [emphasis added], sometimes dramatically so.” Doesn’t this smack of arbitrage? Affirming his belief that “the CEO of any large financial organization must be the Chief Risk Officer as well,” Buffett declared that he had initiated these positions, would continue to monitor them, and would take full responsibility if the company lost profits on these derivatives trades. We discuss later in the chapter how traders (including insurance companies) use options to hedge and speculate on catastrophic risks. Many foundations and endowments, mutual funds, and pension funds actively trade options. Options often underlie financial engineering techniques used by nonfinancial companies, which buy inputs to produce outputs. Sometimes governments use options. Recall the example at the beginning of Chapter 14, in which Lawrence Summers and Zhu Rhongji discussed the feasibility of using put options to defend the value of the Chinese currency and how the US government used warrants to design a highly effective financial rescue package for troubled banks. Options trading strategies adopted by different kinds of traders get regularly mentioned in the financial press and addressed in case studies.

15.3

Profit Diagrams

As pictures say a thousand words, we use profit diagrams as our primary tool for discussing options trading strategies. Recall that profit diagrams place the stock price on the expiration date S(T) on the horizontal (x) axis and the net profits from trades on the vertical (y) axis. The profit or loss is computed by vertically adding up the profits and losses for each security within the trade. Wherever convenient, we show the strike price K, the maximum profit (or loss), and the stock price corresponding

OPTIONS STRATEGIES

to the breakeven point (BEP; or the zero-profit point), where the trader recoups the option premium. Absent market frictions, the zero-sum nature of options trading ensures that the BEP is the same for both the buyer and the writer. Profit diagrams have some limitations. First, you can only draw them for European options. American options have the risk that one or more legs of the strategy may get exercised early and disappear from the diagram. Second, they do not apply before expiration because the time value of the options (see Chapter 5) moves their profits away from those on the expiration date. Third, we must have just one stock as the underlying. You cannot draw profit diagrams for a portfolio of options on different stocks. Still these diagrams are quite useful. If American options remain unexercised until expiration, they have the same payoffs as European options. Although most exchangetraded options in the US are American options, early exercise isn’t very common. The next chapter will demonstrate that options are usually worth more alive (i.e., unexercised) than dead (exercised). Profit diagrams provide a useful snapshot of the possibilities on the option’s maturity date. We focus on equity options. With minor modifications, you can similarly analyze other options, including those on commodities, indexes, and even futures and forwards. For now, we examine simple options trading strategies.

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15.4

Options Strategies

A story titled “Options Trading Grew Up in 2009” (Wall Street Journal, January 4, 2010) reported that many new options traders entered the market in 2009. What attracted these options traders? The article identified several factors. Many investors traded options to better manage their portfolio’s risks after suffering losses. Some adopted complex strategies that “helped them to reduce trading costs and limit the impact of ‘decay,’ which describes the pace at which options lose value as they approach expiration.” To prevent losses, they bought put options or “put spreads,” and to generate income, they tended to sell “covered calls.” This and the next two sections discuss these options trading strategies.

Common Options Price Data Most of our options trading strategy examples will use the following common options pricing data (COP data) for a fictitiously named company, OPSY, which is the Dutch name for options: ■

The current stock price S is $22.50.

■

The time to maturity T is six months.

■

The strike prices K are $17.50, $20, $22.50, and $25.

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TABLE 15.1: European Option Price Data for OPSY Strike Price

Call Price

Put Price

K0 = $17.50

$5.50

$0.10

K1 = 20.00

3.50

0.50

K2 = 22.50

2.00

1.50

K3 = 25.00

1.00

3.00

Stock price S = $22.50; time to maturity T = 6 months; risk-free interest rate r = 5 percent per year.

■

The continuously compounded, risk-free interest rate r is 5 percent per year.

■

The European options prices are given in Table 15.1. These assume a volatility equal to 0.26.

We ignore market imperfections such as transaction costs and also interest earned on the option’s premium. Usually these are small. Of course, you can easily add them to the profit or loss amounts.

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Naked Trades: The Basic Building Blocks The basic option trading strategy building blocks are stand-alone positions containing a single option. Traders refer to these as naked trading strategies (who said options are boring?). Long naked trades are outright purchases because you are not holding an offsetting position in the underlying stock. Short naked trades are also known as uncovered strategies because you are not covering (by reducing the risk of) the trade by simultaneously taking a position in the underlying stock.

EXAMPLE 15.1: Going Long and Selling Short: Stocks and Options Before illustrating the four uncovered positions for European call and put options (with strike prices of $22.50), we draw profit diagrams for long and short stock trades (see Figure 15.1). You can evaluate option trades vis-á-vis these key references. Naked option trading strategies are popular among traders.

Long and Short Stock ■

As stocks are limited liability assets, the maximum loss is the price paid. Long OPSY’s profit diagram is a straight line that emanates from a loss of $22.50 along the vertical axis. It rises at a 45 degree angle, cuts the horizontal axis at the breakeven point of $22.50, and continues upward with infinite profit potential.

■

The short stock’s profit diagram is the mirror image of the long stock across the x axis. An oft-quoted Wall Street saying attributed to the notorious financier Daniel Drew explains the riskiness of short selling: “He who sells what isn’t his’n, buys it back or goes to prison.”

OPTIONS STRATEGIES

Long Call ■

Buying OPSY 22.50 European calls for $2 gives the right to buy the stock for $22.50 at time T. Recall from Chapter 5 that the profit diagram is a horizontal line starting from (a maximum loss of) $2 on the vertical axis. Then, at the strike price of K = $22.50, it shifts upward at a 45 degree angle, breaks even where the stock price S(T) equals the strike price plus call premium (BEP = $24.50), and increases with an unbounded profit potential.

FIGURE 15.1: Profit Diagrams for the Basic Security Blocks (Naked Strategies) Profit

Profit Long stock 1

22.5

1 22.5

0

45°

22.5

0 S(T) Stock price at expiration

–22.5

S(T) 1 1

Short stock

Call and put strike price K = $22.50 Profit

Profit

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Long call

21

Long put 22.5

24.5 2 0 –2

1.5

45° S(T) 22.5 Short call Breakeven point for call = $24.50

■

–1.5 –21

45°

S(T)

21 Short put Breakeven point for put = $21

Applications and uses: - Buy calls to benefit if the stock goes up while limiting losses to the premium if the stock goes down. You can buy OPSY stocks for $22.50 per share. Alternatively, you can buy OPSY 22.50 six-month calls for $2 per share. Each trade has its pros and cons. Action on the call happens during its short life span. By contrast, you can hold on to the stock forever in the hope of a price increase. If OPSY goes down to $15, you lose $7.50 per share but just $2 on the call. If OPSY goes up, you immediately start benefiting on the long stock, but you have to wait until the stock price crosses the breakeven point of $24.50 before profiting on the call.

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- Let’s look at percentage returns. A $7.50 decline is a 33.33 percent loss on the stock investment, but a $2 loss on the premium is a 100 percent loss on the option. If the stock goes up to $30, then there is a 33.33 percent profit on the stock but a (30 – 24.50)/2 = 275 percent profit on the call. The dollar gain or loss is higher for the stock, but the percentage gain or loss is higher for the option. This happens because options are leveraged investments. - Buy calls to speculate on events, corporate developments, and sector and economic performance.

Short Call ■

Selling a call option is also known as shorting or writing a call. A short call’s profit diagram is the mirror image across the x axis of that for a long call.

■

Naked call writing is one of the riskiest strategies, and only seasoned option traders with sufficient financial resources are allowed to employ this strategy.

Long Put ■

A put buyer loses the entire premium of $1.50 if the stock price stays above the strike but starts recouping his investment when S(T) dips below this mark. The breakeven point is at $21, where the $1.50 gain exactly offsets the premium paid. Profit is maximized when S(T) hits zero. Here a worthless stock is sold for $22.50, and deducting the premium arrives at a profit of $21. - Put purchasers bet on the downside but limit their losses in case the bet goes wrong. As with calls, a long put has similar pluses and minuses vis-à-vis a short stock trade.

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Short Put The uncovered put writer has the obligation to buy OPSY for $22.50 on assignment of an exercise notice. A put writer is uncovered if he does not have a short stock position or has not deposited cash equal to the strike to form a cash-secured put. Naked put sellers take losses during market crashes. One of the most notorious examples happened during the Black Monday crash of October 19, 1987. Buoyed by the large stock market gains during the mid-1980s, naked put sellers viewed put premiums as free cash. But when the Dow fell 22.61 percent on this fateful day, many of these put sellers were wiped out. ■

Naked put writers often have two objectives: to receive premium income and to acquire stock at a cost below its market price at some future date.

■

Short put strategies are bullish.

To summarize, naked options trades generate profits that are similar to those created by trading stocks without their downsides. But they are created for a limited time and require regular payment of premiums to maintain the position over long time periods.

HEDGED STRATEGIES

15.5

Hedged Strategies

Hedged strategies combine options and stocks in ways that reduce the overall risk of naked option trades. They are usually covered strategies because they hedge option transactions with the stocks that protect (cover) them.

EXAMPLE 15.2: Hedged and Covered Strategies

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Covered Call Writing (Long Stock plus Short Call) ■

Covered call writing combines a short call with the underlying stock. Figure 15.2 illustrates this 1:1 hedge strategy. - At the origin, the long OPSY trade incurs the maximum loss ($22.50), but the short call retains the entire call premium of $2. The combined profit is – 22.50 + 2 = – $20.50. - As the stock price increases, the short call’s profit graph (a dotted line parallel to the horizontal axis) remains constant at $2, but the long stock’s profit graph (an upward sloping dashed line) rises by a dollar for each dollar increase in S(T). The resulting profit diagram (shown by a solid line) rises at a 45 degree angle, until the point at $22.50. - At the strike price of $22.50, the profit is $2 (from short call) + 0 (from long stock) = $2. For higher values of S(T), the short call’s profit decreases at a negative 45 degree angle, which neutralizes the long stock position’s increase. Consequently, the covered call’s profit diagram is a flat line for all higher values of OPSY.

■

Doesn’t the profit diagram for this concoction look like the diagram for a short put position? The next chapter will explain why this happens.

■

The timing of the trades gives this strategy different names. It’s called a buy–write if one simultaneously buys the stock and sells the call, but it’s an overwrite if one sells the call after purchasing the share.

■

Buy–write’s popularity prompted the CBOE to develop an innovative benchmark for measuring the performance of such strategies. Released in April 2002, the award-winning CBOE S&P 500 BuyWrite Index (BXM) is based on the total returns from hypothetically holding the S&P 500 stock index portfolio and writing a slightly out-of-the-money, one-month maturity call option on the S&P 500 index.

■

Traders who write covered calls generally have two objectives: collecting a premium or hedging a stock trade. By accepting the premium, the writer surrenders the opportunity to benefit from a stock price rise above the exercise price. The call premium gives the writer some cushion against a stock price fall, but he remains vulnerable to losses from a deep decline in the stock. Covered call writing works well in neutral markets, particularly when the volatility is high, which translates into higher call prices.

■

Covered call writing is a more conservative strategy than an outright stock purchase. A director of investments at a pension fund went as far as to comment that covered call writing is one of the most conservative strategies possible.

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FIGURE 15.2: 1:1 Hedges: Covered Call Writing (Buywrite or Overwrite) and Covered Put Buying (Married Put or Protective Put) Profit

Long stock 20.5

Covered call writing

2 0

S(T) 22.5 Short call 1 1 –20.5 –22.5 Profit 21

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Long put Covered put buying 22.5 0 –1.5

S(T) 24 1 1 Long stock

–22.5

■

Even the government views covered call writing as a less risky strategy. You can write covered calls in some tax-advantaged retirement savings accounts. In June 2009, the Securities and Exchange Commission approved allowing employees to sell calls against unexercised holdings of employee

HEDGED STRATEGIES

stock options (which are call options on the stock that are granted by the company as a form of noncash compensation; see “Stock Options Opened for ‘Call Writing,’” Wall Street Journal, June 26, 2009).

Covered Put Buying (Long Stock plus Long Put) ■

Another highly popular strategy, covered put buying has long been used to justify the trading of options: to buy insurance by hedging a long stock with an option. Figure 15.2 illustrates this strategy.

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- At the origin, the long OPSY trade loses $22.50 but the long put position has a gain of $21, giving a profit of – $1.50, which is the maximum loss. - For each dollar increase in S(T), the long put decreases the covered put strategy’s profit by a dollar, giving a profit parallel to the horizontal axis. - When S(T) reaches K = $22.50, the long put’s profit changes and becomes parallel to the x axis, but the long stock position keeps increasing. As a result, the profit rises at a 45 degree angle, signifying boundless profit potential. ■

As with the previous strategy, the timing of the trades gives the strategy different names. It is called a married put (from an old Internal Revenue Service ruling, states the Options Industry Council website) if one simultaneously buys the put and the stock, but a protective put if one buys the put to protect the downside risk of a stock previously purchased.

■

Again, the profit diagram looks like another option trade, a long call position. We will explain this in the context of put–call parity in the next chapter.

■

Traders employ covered put strategies for several reasons: - Establishing a minimum selling price for the stock. No matter how far the stock declines, your loss is limited to the put premium plus the difference, if any, between the stock’s purchase price and the put’s exercise price. In this case, the minimum selling price is (Strike price – Put premium) = 22.50 – 1.50 = $21. - Buy puts to protect unrealized profit in a long stock position. Suppose you bought OPSY when it was trading for $15 and it has now reached $22.50. You can set up a protective put position to hedge your profit against short-term price declines. No matter how far the stock falls, you can always exercise the put and sell the stock at the strike price $22.50. Your profit never falls below [(Strike price – Stock purchase price) – Put premium] = (22.50 – 15) – 1.50 = $6.

Covered Call Buying (Short Stock plus Long Call) Suppose you are bearish and sell OPSY short. This is a high-risk strategy. What’s your protection if the stock rallies instead? One can hedge the risk by buying a call (perhaps an out-of-the-money call like OPSY 25, which has a price of $1).

Covered Put Writing (Short Stock plus Short Put) A less popular strategy, covered put writing tries to generate some income for the short sellers.

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The 1:2 Hedge The previous four strategies considered 1:1 hedges (see Figure 15.2), in which one option was covered by one stock. You can also create hedged strategies by combining unequal numbers of options. A 1:2 hedge combines a long stock with two short calls (see Figure 15.3) or two long puts. The profit diagram looks similar to the diagram of a straddle discussed later in this chapter. These strategies have long been used by dealers who write options; for example, when options are at-the-money, a call writer may buy half a share for each call sold, whereas a put writer may sell half a share for each put sold. The precise quantity of stocks to hedge options trades to remove all of the portfolio’s price risk is given by the stock’s delta, which is discussed in Chapters 17, 18, and 19 and in considerable detail in Chapter 20.

FIGURE 15.3: Long Stock and Two Short Calls (1:2 Hedge) Profit Long stock 4 0

26.5

18.5

S(T)

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22.5 1:2 Hedge

–18.5

Two short calls

–22.5

15.6

Spread Strategies

Oxford University professor John L. Austin asserted during a lecture that although there are many languages in which a double negative makes a positive, no language uses two positives to express a negative statement. Seated in the audience was another eminent philosopher, Columbia University professor Sidney Morgenbesser. He dismissively dissented, “Yeah, yeah,” and the tone of his voice disproved the claim. In finance, we can use a variation of this venerable philosopher’s assertions to describe a spread strategy in which two risky trades combine to form a less risky portfolio. This happens because a spread strategy subtracts “similar” risks, making the end product safer than either of the two originals. Because spreads are two-sided hedges

SPREAD STRATEGIES

that combine two options of the same type but on opposite sides of the market, the exchanges allow spreads to have lower margins than naked trades. A spread gives a smaller profit if the underlying moves in one direction but a tiny loss otherwise. Choice of which option to buy and which to sell determines whether it is a bullish or a bearish spread. Spreads come in three basic types: (1) vertical spreads (also called money, perpendicular, or price spreads) involving options that have different strike prices but expire on the same date; (2) horizontal spreads (or time or calendar spreads) involving options that have the same strike but different maturity dates; and (3) diagonal spreads involving options that differ both in terms of strike price and maturity date. Different maturity dates prevent representing diagonal and horizontal spreads in standard profit diagrams. Instead, traders compute options sensitivities to understand the behavior of such portfolios. Chapter 20 explains these tools.

EXAMPLE 15.3: Bull and Bear Spreads

Bull Spread

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■

You can create a bullish vertical spread by buying an option and simultaneously selling an option of the same type but with a higher strike price, where both options have the same underlying security and the same date of maturity.4 Traders set up bull spreads when they are optimistic about the underlying stock. We create a bull call spread by trading OPSY calls from our COP data, as follows: - Buy the OPSY call with a strike price K1 = $20 for $3.50 and sell a call with K2 = $22.50 for $2 and draw their profit graphs (see Figure 15.4). - If the stock price at expiration is zero, the loss of $3.50 on the first call is partially offset by a $2 gain on the second call, giving a loss of $1.50. - The loss stays constant until S(T) reaches $20. Beyond this, the long call’s profit graph increases at a 45 degree angle, while the short call’s profit graph remains flat. Consequently, the profit graph for the bull spread, which was $1.50 below the horizontal axis, moves up dollar for dollar for each dollar increase in S(T), cuts the x axis at the BEP = (20 + 1.50) = $21.50, and reaches $1 when S(T) reaches $22.50. - Beyond $22.50, the short call position decreases at a negative 45 degree angle. This decline neutralizes the upward rise from the long call, and the net result is a flat line that is $1 above the x axis.

■

This is a debit spread because it costs money to create the spread, as opposed to in a credit spread, where you receive money in the creation process. You create a credit spread when you enter into a bullish put spread by buying a put and simultaneously selling another put with a higher strike price. For example, by buying the OPSY 20 put and selling the OPSY 22.50 put, you create a bull spread that has a maximum loss of $1.50, a BEP at $21.50, and a maximum profit of $1—a profit graph that happens to be identical to the graph of our previous bull call spread (see Figure 15.4).

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FIGURE 15.4: Vertical Bull Spread with Calls Profit

20

2 1 0 –1.5 –3.5

Long call (K1 = 20)

21.5

Bullish call spread S(T) 22.5

Short call (K2 = 22.5)

FIGURE 15.5: Vertical Bear Spread with Puts Profit

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21

Long put (K2 = 22.5) 1.5 0.5 0 –1 –1.5

Bearish put spread

21.5

S(T) 22.5 20

–19.5

Short put (K1 = 20)

Bear Spread ■

A bearish vertical spread is created by buying an option and simultaneously selling an option of the same type, but with a lower strike price, where both options have the same underlying security and maturity date. We create a bear put spread by trading OPSY puts as follows: - Sell an OPSY put with strike price K1 = $20 for $0.50 and buy a put with K2 = $22.50 for $1.50 and draw their profit diagram (see Figure 15.5).

SPREAD STRATEGIES

323

- At the origin, the long OPSY put trade gains $21, but the short put position loses $19.50, giving a payoff of $1.50, which is the maximum profit. For higher values of S(T), the profit is a flat line because the long put declines but the short put rises, and the two effects cancel each other. - The profit stays constant until S(T) reaches $20. Beyond this point, the long put’s profit continues to decline at a 45 degree angle, whereas that for the short put remains flat. The profit graph for the bear spread, which was $1.50 above the horizontal axis, decreases dollar for dollar for each dollar increase in S(T), cuts the x axis at the BEP (20 + 1.50) = $21.50, continues declining, and reaches –$1 when S(T) touches $22.50. Beyond this, the long put position also becomes a flat line, and the payoff becomes a line parallel to the x axis. - Notice that the bear put spread of Figure 15.5 is a mirror image of the bull put spread of Figure 15.4 (which is the same as a bullish call spread) across the horizontal axis. This is no surprise because the two spreads involve identical options that are traded in opposite directions. - You can similarly create a bear call spread by reversing the trades that led to the creation of the bull call spread. You can create profit diagrams that look like vertical spreads by trading the underlying asset and options. This is demonstrated in Extension 15.1.

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4 For a bullish horizontal spread, buy an option and sell another option of the same type and strike but with shorter life. For a bullish diagonal call spread, buy an option with a lower strike price and longer time to maturity than the one that is sold. Do the opposite for bearish spreads.

A spread can be used as a starting point for understanding more complex strategies. For example, you can begin with a vertical spread and tinker with it by adding more options of the same type. This leads to ratio spreads, butterfly spreads, condors, and other options trading strategies, some of which are popular and others obscure. Extension 15.1 shows how to do this. Moreover, one can find options spreads embedded in other contexts. For example, a reinsurance contract is equivalent to a call spread. This connection is explored in Extension 15.2.

EXTENSION 15.1: Collars, Ratio Spreads, Butterflies, and Condors Consider a simple example. Suppose the portfolio your boss manages tries to mimic the S&P 500 index, but it allows some tinkering. You advise buying some puts on the index with strike K1 with cost $p per option and financing this by selling calls with the same price $c but a higher strike price K2 . Ext. 15.1 Fig. 1 shows that the profit graph of this zero-cost collar (long stock, long put[s], and short call[s], where the total cost of the calls equals that of the puts) is a spread. The strategy has its trade-off. You are pleased if the underlying declines and the put protection works but terribly glum if the underlying rises and the gains are surrendered to the call buyer. Collars are similar to zerocost collars but do not require the cash flows from the calls and puts to negate each other. You can establish a collar on a long stock position by adding a covered call and a protective put.

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You can create butterfly spreads and condors by modifying a vertical spread. We demonstrate this by first drawing a vertical bull spread with our COP data. Then we create a whole range of spread strategies by adding options of the same type (calls with calls or puts with puts), but with different strike prices.

EXT. 15.1 FIG. 1: A Zero-Cost Collar Profit

Long stock

Long put K1

Zero-cost collar

c 0 –p

S(T) K2

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Short call

Here call price c = put price p

EXT. 15.1 EX. 1: Examples of a Call Spread, a Ratio Spread, Butterflies, and Condors

Bull Call Spread One can create a bull call spread similar to the one in Example 15.3 by buying one OPSY 17.50 call for $5.50 and selling one OPSY 20 call for $3.50. The profit diagram is initially a flat line with a loss of $2, rises at a 45 degree angle as the stock price goes from $17.50 to $20, breaks even at $19.50, and again becomes a flat line at a profit of $0.50 when S(T) exceeds $20 (see Ext. 15.1 Fig. 2).

Ratio Spread ■

Now sell another OPSY call with strike price $20. By selling two calls, you collect a premium of $3.50 × 2 = $7, and the profit diagram declines by $2 for each dollar increase in S(T) beyond $20. This creates a ratio spread, which is defined as an options trading strategy in which the number of calls (or puts) purchased is

SPREAD STRATEGIES

325

different from the number of calls (or puts) sold. The profit graph for this one-by-two call spread is a flat line emanating from (– 5.50 + 7) = $1.50 on the vertical axis, which then rises at 45 degrees when S(T) exceeds $17.50 and attains the peak value of (1.50 + 2.50) = $4 when S(T) is $20 (see Ext. 15.1 Fig. 2).5 Beyond this, the long call moves the graph up, but the two short calls drag it down twice as fast, resulting in a line that declines at a negative 45 degree angle.

Butterfly Spread ■

A butterfly spread is created by trading four options of the same type (all calls or all puts) with three different strike prices: two options with extreme strike prices are bought (written) and two options are written (bought) with the middle strike price (see Ext. 15.1 Fig. 2).

■

A butterfly spread is a two-sided hedge that bets on the volatility of the underlying security: a long butterfly spread makes a small profit if the volatility is low and a small loss otherwise. The profit graph for this long butterfly call spread starts at a maximum loss of (–5.50 + 7 – 2) = – $0.50 along the vertical axis. Initially a line parallel to the horizontal axis, the profit graph starts increasing when S(T) exceeds $17.50, breaks even at $18, attains a maximum value of $2 when S(T) is $20, declines and cuts the x axis again at $22, falls to – $0.50– $0.50 at $22.50, and again becomes a line parallel to the x axis for all higher values of S(T).

■

A butterfly spread is so named because the profit diagram looks like a butterfly’s wings. If you reverse the preceding trades, you will create a short butterfly spread. You can also create butterfly spreads with puts.

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Condor ■

Thinking that OPSY might fluctuate more than anticipated, you decide to expand the region over which you want to profit. You modify the butterfly spread by (1) keeping a long call with K0 = $17.50 as before, (2) (instead of two short calls at the same strike) selling one call at the next higher strike price K1 = $20 and another at K2 = $22.50, and finally, (3) buying a call with the uppermost strike K3 = $25 (see Ext. 15.1 Fig. 2). This creates a condor spread, which has four options with four strike prices: long two options with extreme strikes and short two options with strike prices in the middle, and vice versa. The wings of the condor, the largest flying land bird in the western hemisphere, inspire the name.

■

Our condor’s profit diagram begins on the y axis at (–5.50 + 3.50 + 2 – 1) = – $1, the amount obtained by adding up the premiums. Initially, it’s a line parallel to the x axis because the profits from the calls are flat at this point. When S(T) goes beyond 17.50, the call with the lowest strike becomes active and increases the payoff. The profit graph reaches a maximum value of $1.50 when S(T) is 20. It becomes a flat line for S(T) lying between $20 and $22.50 because the call with a strike of K0 = $17.5 pulls it up but the call with K1 = $20 pushes it down. Beyond $22.50, the third call (with K2 = $22.50) kicks in, and the profit graph falls. Finally, when S(T) exceeds $25, the fourth option (with K3 = $25) also becomes active and increases the payoff: with two calls pulling it up and two pushing it down, it again becomes a flat line. You can also create a condor with puts.

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EXT. 15.1 FIG. 2: Bull Call Spread, Ratio Spread, Butterfly, and Condor Profit Long call (K0 = 17.5)

17.5 3.5 0.5 0 –2 –5.5

Bullish call spread S(T) Short call (K1 = 20) 20

Profit 7 4 1.5 0 –5.5

Long call (K0 = 17.5)

17.5

S(T)

1 20

1 Call ratio spread

2 1

Two short calls (K1 = 20)

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Profit Long call (K0 = 17.5) Long call (K2 = 22.5)

7 17.5 2 0 –0.5 –2 –5.5

22.5

20

S(T) Butterfly call spread

Two short calls (K1 = 20) Profit

17.5

25

1.5 0 –1

S(T) Condor spread 20 22.5

SPREAD STRATEGIES

327

5 A call ratio spread typically involves buying calls and selling a greater number of calls at a higher strike price. If you sell calls and buy a greater number of calls at a higher strike price, then you create a call backspread.

EXTENSION 15.2: Reinsurance Contracts and Call Spreads Reinsurance contracts on a loss causing event (e. g. hurricanes) are equivalent to a call spread in which the call options are written on the losses of the insured event. To understand this equivalence, consider the following example.

Risks Facing the Insurance Industry

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Consider some of the risks in the insurance industry. ■

Dearth of natural hedges. Insurance is a high risk business that cannot be hedged using traded financial securities. Insurance companies employ applied mathematics and statisticians called actuaries, who calculate the likelihood of loss events and fair premiums for insuring these risks.

■

Diversification and the law of averages. Diversification is the industry’s key mantra. Insurers try to manage risks by diversifying their losses across a large number of policyholders. The business survives on the law of large numbers, which says that as more and more independent and identically distributed observations are collected, the average gets closer and closer to the mean value for the population. Although no one can foretell whether any particular insured entity will suffer a loss, actuaries can estimate with reasonable accuracy the expected losses in a large population. They compute fair premiums based on these expected losses. Insurance companies lose money when the realized losses exceed the expected losses.

■

Reinsurance. For catastrophic events like major earthquakes and hurricanes, the law of large numbers does not apply because these events are not independent and identically distributed across the population. Insurance companies hedge their losses on such catastrophies by buying their own insurance. This is called reinsurance. We discuss reinsurance after the following example.

EXT. 15.2 EX. 1: Hedging Hurricane Risk ■

Catastrophic Insurance Corporation (CatIns Corp., a fictitious name) sells homeowner’s insurance contracts to one hundred thousand homes along the shores of the Gulf of Mexico in the states of Florida, Alabama, Mississippi, Louisiana, and Texas. The contract provides protection against hurricane risk, which is the risk of losses coming from powerful storms, and it has the following features: - Annual premium $15,000. - Fixed-amount deductible $10,000. - Maximum coverage amount $300,000 (which is less than the value of each home). - Contract pays for losses from one hurricane in a given year. For simplicity, assume that if a hurricane hits, it does the same dollar damage to all homes covered by the insurance policy. These losses manifest the

328

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inability of the insurance company to diversify these losses in a large pool of the insured. Also assume that underwriting costs and investment income net out to zero. ■

We draw the profits for CatIns in a modified profit diagram in which “profit per contract” for the insurance company is plotted (as before) along the vertical axis and the underlying “loss per home” (instead of underlying asset price) is depicted along the horizontal axis. Of course, you can draw the profits in a usual profit diagram. However, this representation is more natural in the context of insurance, where losses are key. We determine the profits as follows (see Ext. 15.2 Fig. 1; both variables are graphed in thousands of dollars).

EXT. 15.2 FIG. 1: Call Spread Diagram Profit per contract (thousands of dollars) 15 12 0

25 10

–78

100 22

300 Loss per home (thousands of dollars) CatIns’s profits (with reinsurance)

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CatIns’s profits (no reinsurance) –275

No Hurricane CatIns keeps the entire premium of $15,000, which is shown as 15 in the diagram. For one hundred thousand homes, this turns out to be $1.5 billion.

If Hurricane Damage Happens For the first $10,000 in losses, the company pays nothing. The profits (shown by a shaded line) remains parallel to the horizontal axis at $15,000. For higher losses, the company pays a dollar for each dollar increase in loss. The profit line declines at a 45 degree angle and cuts the x axis when the total loss reaches $25,000, shown as 25 in the diagram. Beyond this, the profit declines until the total loss on each home reaches the maximum loss of $300,000, where CatIns pays $275,000. This is a profit of $27.5 billion, a staggering sum considering that the company earned only $1.5 billion in premiums.

Insurer Buys Reinsurance CatIns can buy reinsurance that pays for losses to a home over $100,000 for a premium of $3,000 payable to a reinsurance company. Then, the new payoff line emanates from the y axis at (15,000 – 3,000) = $12,000, stays

COMBINATION STRATEGIES

329

parallel to the x axis until “loss per home” reaches $10,000, goes down thereafter at a 45 degree angle, cuts the horizontal axis at $22,000, reaches a maximum loss of (100,000 – 22,000) = $78,000 when “loss per home” reaches $100,000, and again becomes parallel to the x axis because the reinsurance company pays for the higher losses.

Equivalence to a Call Spread This loss diagram shows that the profits to CatIns with reinsurance are equivalent to a call spread on the hurricane losses. The company makes a limited profit if the hurricane’s losses are mild, and the company incurs losses that are capped at $7.8 billion when large hurricanes hit homeowners.

Reinsurance Markets

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Reinsurance has a long history. One of the earliest known reinsurance contract agreements was written in Latin and signed by Italian merchants on July 12, 1370. Reinsurance helps insurance companies manage tail risk, which refers to the risk of occurrence of infrequent events that cause large losses when they occur. Tail risk is so named because these large loss events lie on the extreme end (“tail end”) of a probability distribution of such loss events. As illustrated by the example, an insurance company’s profit diagram under a typical reinsurance contract is equivalent to a call option spread. The call options are written with a one-year maturity on the aggregate losses realized by the insurance company, where the strikes correspond to the initial deductible and the payment cap. Given the call spread analogy, reinsurance contracts can be priced using the option pricing methodologies presented in Chapters 17–20.

15.7

Combination Strategies

“A double privilege pays a profit, no matter which way the market goes, and costs $212.50,” declared an advertisement in 1875 in the book Secrets of Success in Wall Street by Tumbridge and Co., Bankers and Brokers, which had its office on 2 Wall Street, New York City at that time. An ancestor to today’s booklets that aim at informing and attracting traders to options, this forty-eight-page volume describes the workings of the New York Stock Exchange and the over-the-counter options market. The traded contracts went by the name stock privileges and were essentially American options: calls, puts, spreads, and straddles that were customarily written on one hundred shares of stock and matured in thirty days. Tumbridge charged $100 as the premium for a call or a put and a $6.25 broker’s commissions for each leg of the trade. Double privilege referred to spreads and straddles, which were created by trading two options, and hence charged two commissions. Straddles and strangles are examples of combination strategies that combine options of different types on the same underlying stock and expiring on the same date, where the options are either both purchased or both written. For example, buying a call and a put with the same strike price creates a straddle (called a putto-call strategy in London); if their strike prices are different, you get a strangle.

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Straddles and strangles are volatility plays. Options traders use them to bet on risky future events affecting a company that can impact the stock price in either a positive or negative direction. Traders sometimes write them when they expect neutral markets.

EXAMPLE 15.4: Straddles and Strangles

Straddle ■

You buy a straddle (a long straddle or a bottom straddle) by buying a call and a put as follows: - Buy OPSY 22.50 options, the call for $2 and the put for $1.50, and draw the profit diagram (see Figure 15.6).

FIGURE 15.6: Straddles and Strangles—Examples of Combination Strategies Profit 21 19

Long put Long call Straddle

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22.5 19

0 –1.5 –2 –3.5

26 S(T)

Profit 19.5 17.5

Long put Long call Strangle

20 22.5 0 –0.5 –2 –2.5

17.5

25 S(T)

SUMMARY

- If the stock price at expiration is zero, the long put makes $21, but the long call loses $2, giving a gain of $19. As S(T) increases, the long call’s payoff remains parallel to the horizontal axis, but the long put drags the straddle’s payoff down at 45 degrees. The payoff cuts the x axis at the breakeven point $19, keeps declining, and reaches the nadir—a loss of $3.50. - For values of S(T) higher than $22.50, the long put’s payoff becomes parallel to the x axis, but the long call pulls the straddle’s payoff up at 45 degrees, which cuts the x axis again at the BEP = (22.50 + 2 + 1.50) = $26. For higher values of S(T), the payoff increases at 45 degrees, reflecting infinite profit potential. ■

If you reverse the preceding trades and sell a call and put, then you sell a straddle (establish a top straddle or a short straddle), a bet on low future volatility whose payoff is the mirror image of the bottom straddle across the horizontal axis.

Strangle

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■

A long strangle is created by buying options with different strike prices, where the strike price of the put is lower than the strike price of the call. Though it’s a volatility bet like a straddle, a strangle does have some differences: - Buy the OPSY 22.50 call for $2, and buy the OPSY 20 put for $0.50. Plot these two strategies in a profit diagram (see Figure 15.7). - At the origin, the profits for the strangle is 19.50 – 2 = $17.50. For higher values of S(T), it declines at a 45 degree angle until it reaches –$2.50 when S(T) is $20. For the stock price lying between $20 and $22.50, the profit is flat because both the call’s and put’s profits are parallel to the horizontal axis. Beyond $22.50, the long put’s profit stays flat, but the long call pulls up the strangle’s profit, which cuts the x axis at $25 and keeps increasing at a 45 degree angle, indicating an unlimited profit potential.

15.8

Summary

1. Options trading strategies’ performances are drawn in profit diagrams for European options with a single stock as the underlying. The basic building blocks for options trading are simple naked (or uncovered) strategies that are stand-alone positions buying or selling a single call or put option. 2. Hedged strategies combine options and stocks in a way that dampens the overall risk as compared to a naked option trade. They are usually covered strategies because they tend to back the option transaction with a stock that protects (or covers or collateralizes) it. Hedged strategies include covered call writing (long stock plus short call) and covered put buying (long stock plus long put). 3. Spread strategies combine several options of the same type (either both calls or both puts) on the same underlying but on different sides of the market. These two-sided hedges give a small profit if the underlying moves in one direction but suffer a small loss otherwise.

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4. Spreads come in three basic types: (1) vertical spreads involve options that have different strike prices but expire on the same date, (2) horizontal spreads involve options that have the same strike but different maturity dates, and (3) diagonal spreads involve options that differ both in terms of strike price and maturity date. 5. Several other categories of spread strategies can be created by adding options of the same type, with the same or different strike prices, to a spread strategy: - A ratio spread has the number of calls (or puts) bought different from the number of calls (or puts) sold. - A butterfly spread has four options of the same type (all calls or all puts) with three different strike prices: buy (sell) two options with extreme strike prices and write (purchase) two options with a middle strike price. - A condor spread has four options with four strike prices: purchase (write) the two extreme options and sell (buy) two options with strike prices in the middle. 6. Combination strategies combine options of different types on the same underlying stock and expiring on the same date, where both options are either purchased or written. A straddle combines a call and a put with the same strike price. A strangle combines a call and a put with different strike prices.

15.9

Cases

Cephalon Inc. (Harvard Business School Case 298116-PDF-ENG). The case intro-

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duces students to the use of equity derivatives as part of a risk-management strategy, examines the application of cash flow hedging in a corporate context, and explains the pricing of a derivative security with large jump risk. National Insurance Corp. (Harvard Business School Case 296036-PDF-ENG).

The case considers whether a major reinsurer should use exchange-traded insurance derivatives for managing catastrophic risk. Compensation at Level 3 Communications (Harvard

Business School Case 202084-PDF-ENG). The case analyzes a compensation plan that rewards managers for the firm’s performance only if the firm’s stock price movement exceeds a benchmark.

15.10 Questions and Problems The next eleven questions are based on the following data for a fictitiously named company, OOPS. ■

Current stock price S is $22.

■

Time to maturity T is six months.

■

Continuously compounded, risk-free interest rate r is 5 percent per year.

■

European options prices are given in the following table:

QUESTIONS AND PROBLEMS

Strike Price

Call Price

Put Price

K1 = $17.50

$5

$0.05

K2 = 20

3

0.75

K3 = 22.50

1.75

1.75

K4 = 25

0.75

3.50

15.1. Use options with strike K2 = $20.00: a. Draw a long call profit diagram. b. Draw a short call profit diagram. c. Draw a long put profit diagram. d. Draw a short put profit diagram. 15.2. Use options with strike K2 = $20.00: a. Draw a covered call (long stock plus short call) profit diagram. b. Give a reason why a trader might want to hold a covered call position. c. Explain the difference between a buy–write and an over-write strategy. 15.3. Use options with strike K2 = $20.00: a. Draw a covered put (long stock plus long put) profit diagram.

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b. Give a reason why a trader might want to hold a covered put position. c. Explain the difference between a married put and a protective put. 15.4. Use options with strike K3 = $22.50: a. Draw the profit diagram for a long call and a short put position for options

with the same strike and maturity date. b. Draw the profit diagram for a long stock and two short calls. 15.5 Using options with strike price K3 = $22.50, draw the profit diagram for a

long call, a short stock, and a short put position for options with the same strike and maturity date. What other investment has this profit diagram? 15.6. a. What is a bullish vertical spread? b. Draw a bullish vertical spread by trading put options with strike prices K2 =

$20 and K4 = $25. 15.7. a. What is a bearish vertical spread? b. Draw a bearish vertical spread by trading call options with strike prices

K1 = $17.50 and K3 = $22.50. 15.8 Draw a butterfly spread by going long calls with strike prices K1 = $17.50 and

K3 = $22.50 and selling short two calls with a strike price in the middle.

333

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15.9 Draw a condor spread by going long calls with strike prices K1 = $17.50 and

K4 = $25 and selling short two calls with each strike price in the middle. 15.10. a. What is the aim of a long (or bottom) straddle strategy? b. Create a long straddle by buying a call and a put with strike price K3 =

$22.50. 15.11. a. What is the aim of a short (or top) strangle strategy? b. Create a short strangle by writing a call with strike price K3 = $22.50 and

a put with strike price K1 = $20. The next five questions are based on the following options price data for Tel Tales Corporations (fictitious name), where the options expire on the same date in May. Draw profit diagrams in each case, clearly showing the stock price corresponding to zero profit, the maximum profit and loss, and so on. Stock

Strike (K)

Call Price

Put Price

$29

$25

$5

$1

30

2

3

35

1

6

15.12 A 3:1 reverse hedge (buy three May 30 calls and short the stock). Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

15.13 A bullish spread (long call with strike price of $25 and short call with strike

price of $30). 15.14 A butterfly spread (long put with strike prices $25 and $35 and short two puts

in the middle). 15.15 A strangle (buy call with K = $30 and buy put with K = $25). 15.16 A straddle (buy a call and a put with K = $30). 15.17 Goldminers Inc. mines and refines ore and sells pure gold in the global market.

To raise funds, it sells a derivative security whose payoff is as follows: ■

Part of the security is a zero-coupon bond (which is sold at a discount and makes no interest payments) that pays a principal of $1,000 at maturity T.

■

Goldminers also pays an additional amount that is indexed to gold’s price (per ounce) at maturity S(T): 0 if S (T) ≤ $1,350 $30 [S (T) − 1,350] if $1,350 < S (T) ≤ $1,400 $1,500 if $1,400 < S (T) Analyze this derivative as a combination of bond and put options.

QUESTIONS AND PROBLEMS

15.18 a. What is a collar in the options market? b. How would you create a zero-cost collar? c. Why might a copper manufacturer find it useful to employ this strategy? 15.19 An insurance company has insured oil fields in the Middle East. Next, it

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purchases reinsurance to manage its “tail risk.” How can the reinsurance company hedge some of its risks by trading derivatives?

335

16 Option Relations 16.1 Introduction

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16.2 A Graphical Approach to Put–Call Parity 16.3 Put–Call Parity for European Options 16.4 Market Imperfections

16.5 Options Price Restrictions EXTENSION 16.3 The Superglue Argument

16.6 Early Exercise of American Options No Dividends Dividends and Early Exercise

EXTENSION 16.1 Put–Call Parity in Imperfect Markets

16.7 Summary

EXTENSION 16.2 Dividends and American Options

16.8 Cases 16.9 Questions and Problems

A GRAPHICAL APPROACH TO PUT–CA LL PARITY

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16.1

Introduction

Eager to find how the options on Palm Inc.’s stock were performing on their inaugural trading day, he switched on his computer, entered the Internet, and logged on to his online brokerage account. Like Alice in Lewis Carroll’s children’s classic Alice’s Adventures in Wonderland, he grew “curiouser and curiouser” and could hardly believe his eyes. Market-traded put options were trading $4 higher than the arbitrage-free price predicted by put–call parity. Filled with enthusiasm, he established a game plan based on the textbook prescription to exploit the mispricing, but he stumbled at the outset. He could not find Palm shares to borrow and sell short, a critical step in constructing the arbitrage. He (one of us) did not make millions that morning but acquired knowledge and, most important, a story to share with you! Later in this chapter, we will discuss the Palm arbitrage in greater detail. The last chapter analyzed various options trading strategies using their profit diagrams on the maturity date. Determining an option’s price prior to expiration is a far more difficult task. Although it’s easy to find the forward price using the no-arbitrage argument because of the forward’s linear payoff, the nonlinear option payoffs complicate the argument. In fact, to get an exact value, one needs to assume an evolution for the underlying stock price (which indicates how the stock price evolves through time), the most popular being a lognormal distribution. This is studied in subsequent chapters. But what can you learn about options properties without assuming such an evolution? This is the topic studied in this chapter. Here we study three different categories of options price relations. First, we establish put–call parity for European options. This fundamental relation stitches together the stock, a bond, and call and put options. Put–call parity is not new: financier Russell Sage used it to circumvent New York State usury laws in the late 1800s. Next, we examine some restrictions that options prices must satisfy under the no-arbitrage assumption. These provide the key steps to our final topic: the early exercise of American options. As early exercise features are embedded in various other financial securities, it is important to understand this topic well. Throughout this chapter, we maintain our standard assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no intermediate cash flows (such as cash dividends), and no arbitrage. However, we will occasionally need to relax the assumption of “no intermediate cash flows” below.

16.2

A Graphical Approach to Put–Call Parity

In The Put-and-Call, published in London in 1902, Leonard R. Higgins describes the workings of London’s over-the-counter options market. In this book, he makes the following intriguing observation: that “a put can be turned into a call by buying all the stock” and that “a call can be turned into a put by selling all the stock.” As observed by Higgins, a clever arrangement of stocks and European options establishes put–call parity (PCP). Initially, we illustrate this result with payoff diagrams in the next example, and later by other methods.

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EXAMPLE 16.1: Establishing Put–Call–Parity by Payoff Diagrams (Replicating a Call) ■

We can replicate a European call option by trading the underlying stock, a European put option, and zero-coupon bonds. The options on the stock need to have a common strike price and a common expiration date. For simplicity we use the common options pricing data (COP data) from Chapter 15.

■

First, buy an OPSY 20 put option whose time T payoff diagram is drawn in Figure 16.1A. Next, buy one OPSY stock and place its payoff diagram below that of the put. Now create a third payoff diagram that sums the first two payoffs for each value of S(T). This payoff graph for the long put plus the stock starts at a value of K = $20 on the vertical axis. When S(T) exceeds $20, the graph rises at a 45 degree angle to the x axis.

■

For convenience, we redraw this portfolio payoff in Figure 16.1B. Draw another diagram underneath this depicting the payoff from shorting a bond with a future liability of $20. The present value of this cash flow today is $20B, where B is today’s price of a zero-coupon bond that pays a dollar after six months. The short bond’s payoff is a line parallel to the x axis at – $20.

■

Now sum the payoffs of the previous positions (Long put + Long stock) and short bonds. The portfolio (Long put + Long stock + Short bonds) replicates the payoff to a long call option with strike price K.

■

As the time T payoffs are the same, the law of one price equates the value of the two portfolios today:

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Long call = Long put + Long stock + Short bond

(16.1)

Expression (16.1) is put-call parity (PCP) for European options. We can view the left side, the call, as a market-traded asset and the portfolio on the right side as a synthetic call (a portfolio that synthetically constructs the traded call’s payoffs).

16.3

Put–Call Parity for European Options

We follow the data–model–arbitrage approach to establish put-call parity (PCP) for European options. As in the previous section, we consider a market-traded call and construct a synthetic call with identical future payoffs. By the law of one price, the traded and the synthetic call must have the same value today. This yields the PCP relation. In this example, instead of the law of one price, we use the equivalent nothing comes from nothing principle instead.

PUT–CALL PARITY FOR EUROPEAN OPTIONS

FIGURE 16.1A: Step One (PCP by Graphs): Long Put plus Long Stock Payoff 20

Long put

FIGURE 16.1B: Step Two (PCP by Graphs): (Long Put plus Long Stock) plus Short Bonds gives a Long Call Payoff Long put + Long stock 20

0 K = 20

S(T) 0

Payoff Long stock

20

S(T)

Payoff 0

S(T)

0 S(T)

Short bonds

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–20 Payoff Long put + Long stock

Payoff

20 Long call 0

0 20

S(T)

K = 20

S(T)

EXAMPLE 16.2: Put–Call Parity for European Options

The Data ■

Consider the COP data of section 15.4 of Chapter 15. - OPSY’s stock price S is $22.50 at time 0. - The European options on OPSY have a common strike price K = $20 and mature in T = 6 months. - Today’s put price p is $0.50. The call price c is to be determined.

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- The continuously compounded risk-free interest rate r is 5 percent per year. Today’s price of a zero-coupon bond paying $1 at time T is B = e−rT = $0.9753. - No dividends are paid on OPSY’s stock over the option’s life. ■

All trades are recorded in an arbitrage table. The first column of Table 16.1 gives the trade description and the second column records today’s cash flows. The expiration date payoffs are presented in the last two columns. The third column reports the payoffs when S(T) is less than or equal to $20, and the last column gives the payoffs when it’s greater than $20.

■

We first implement a stock purchase plan portfolio. - Buy the stock in the spot market for $22.50. The cash flow is recorded as −22.50 in the second column. The stock is worth S(T) after six months. Jot down S(T) in both the third and fourth columns. - Tap the bond market to borrow the present value of the strike price K = $20 by shorting zero coupon bonds. This cash flow reduces today’s payment. Record 0.9753 × 20 today and the liability -20 in the last two columns.

■

Next, using the options market, we create a synthetic short stock portfolio. - Sell one OPSY 20 call. Record this trade as +c in the second column. The call expires worthless if S(T) is less than or equal to $20. For higher values of S(T), the future payoff is a liability. Surrender the stock worth S(T) and receive $20 in return, for a cash flow of [20 − S(T)].

TABLE 16.1: Arbitrage Table for Put–Call Parity

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Portfolio

Today (Time 0) Cash Flow

Expiration Date (Time T) Cash Flow S (T) ≤ 20

20 < S (T)

Stock purchase plan Long stock

−22.50

S(T)

S(T)

Short bonds to borrow the present value of strike (K = $20, B = $0.9753)

0.9753 × 20

−20

−20

Short call

c

0

20 − S(T)

Long put

−0.50

20 − S(T)

0

Cash flows

−22.50 + 0.9753 × 20 + c −0.50

0

0

Synthetic short stock

- Buy an OPSY 20 put. The premium p = $0.50 is written as a negative cash flow today. If S(T) is less than or equal to $20, then the put holder receives $20 but has to surrender the stock worth S(T). Record the put payoff as [20 − S(T)] in this case. The put expires worthless when S(T) exceeds $20.

PUT–CALL PARITY FOR EUROPEAN OPTIONS

■

The first two trades (Long stock + Short bond) have the payoff [S(T) − 20] at expiration. The last two trades (Short call + Long put) create an equal but opposite payoff on the expiration date.

■

The final row of Table 16.1 considers net cash flows. The portfolio’s payoff is always zero on the expiration date.

■

The portfolio of stocks, bonds, and options created has a zero value for sure in the future. To prevent arbitrage, using the nothing comes from nothing principle, it must have zero value today. Consequently, −22.50 + 0.9753 × 20 + c − 0.50 = 0 or c = $3.49 This is the European call option’s arbitrage-free price.

The Model ■

Use symbols to generalize. Replace $22.50 with S, $0.9753 × 20 with B × K, and $0.50 with p to get the PCP for European options: − S + BK + c − p = 0

(16.2)

Arbitrage Profits

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■

A violation of PCP leads to arbitrage. Suppose you find an errant trader who quotes a call price of $4. As this is higher than the arbitrage-free price from PCP, create the portfolio in Table 16.1: buy the stock, short the bonds, buy the put, and sell the relatively overpriced call. This gives the cash flows −S + BK + c − p = −22.50 + 0.9753 × 20 + 4 − 0.50 = $0.51 as your immediate arbitrage profit.

■

If the trader quotes $3 instead, reverse the trades and make $0.49 as an arbitrage profit.

We use expression (16.2) to formally state our result.

RESULT 16.1 Put–Call Parity for European Options c = p + S − BK

(16.3)

where c and p are today’s prices of European calls and puts, respectively, that have strike price K and expiration date T, B is today’s price of a zero-coupon bond that pays a dollar at time T, and S is today’s price of the underlying stock.

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PCP is a useful tool for understanding how to use options in forming trading strategies. We start with the put because it is the easiest option to understand. Indeed, a put option is like an insurance policy because it insures the value of the underlying stock at its strike price. With this insight, we can rewrite PCP as Call = Insurance + Stock + Borrowing

(16.4)

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We see that a call option is equivalent to buying the stock on margin (by borrowing) and protecting the stock purchase with an insurance policy. This intuition can help us to understand how to trade on information using options. For example, suppose that you are the only person in the market who knows that the stock is going to rise above the strike price. (Note: this is a thought experiment—we are not considering issues like whether trading on inside information will attract jail time!) What is the best way to trade on this information: buying a call? No. This is the wrong answer because by buying the call, you are also paying for the put, which is insurance on the stock that you do not need. Recall that you are certain that the stock price will rise. In this case, the optimal strategy is to buy the call and sell the put. Using PCP again, we also see that this is equivalent to buying the stock and financing the purchase with borrowed cash, which creates leverage (called gearing in Europe). Now suppose that you expect that the stock is likely to increase but you are unsure. In this case, is buying the call the best strategy? Maybe. The answer really depends on your risk tolerance. Pay for the put if you like the downside insurance protection but not otherwise.

16.4

Market Imperfections

Market imperfections introduce difficulties, but they may also create opportunities. Merton Miller’s observation that financial securities can help overcome rules and regulations becomes pertinent. Options and PCP can provide tools to work around these imperfections. In the article “The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage,” Michael Knoll (2008) cited ancient examples of merchants using PCP to get around restrictions on interest payments. For example, two millennia back, Israelite financiers used PCP in this way. English financiers did the same five hundred years ago, and interestingly, the transaction they devised led to the development of the modern mortgage. These examples illustrate tax or regulatory arbitrage, which involves unpackaging and rebundling cash flows with the aim of profiting from otherwise prohibited transactions. Extension 16.1 discusses an example of PCP in action. One can tinker with PCP to generate two extensions: (1) to accommodate dividends and (2) for American options. These extensions are shown in Extension 16.1 and 16.2.

MARKET IMPERFECTIONS

343

EXTENSION 16.1: Put–Call Parity in Imperfect Markets

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PCP Violations for Palm Shares Sometimes PCP fails because of market imperfections. This happened with Palm Inc. options in an incident discussed at the chapter’s start. On March 2, 2000, 3Com did an equity carve-out of its subsidiary Palm, which had made the world’s first successful handheld computer, the Palm Pilot, in 1992. Accordingly, 3Com sold 5 percent of Palm shares to the general public in an initial public offering and declared that later in the year, it would distribute the remaining shares in a spin-off to its existing shareholders (who would get 1.525 shares of Palm for every share of 3Com they held). The mispricing that resulted in the equity and options markets got extensive press coverage and academic study (see Lamont and Thaler 2003). A hot company during the days when tech stocks led the dotcom stock market craze, Palm had an unbelievable initial public offering (IPO). Its stock closed at the end of the first day’s trading at $95.06 per share. This implied that 3Com’s stock price, which would equal 1.5 Palm shares plus the value of 3Com’s other assets and businesses was $63 per share, or – $22 billion. The mishmash confused options traders. They adopted strategies like writing call options and short selling 3Com stock to buy Palm stock, whose market capitalization then exceeded the parent company’s valuation! “There is absolutely no rationale behind it,” interjected a trader. Predictably, Palm started declining after the IPO.1 When options began trading on March 16, 2000, Palm stock traded around $55. For options maturing in one month, the at-the-money (strike price K = $55) call price c was $5 and the put price p was $9. These numbers are illustrative. The bid/ask spread was large, and the market was in a state of confusion, making it hard to find accurate prices. A zero-coupon bond price paying $1 after one month was worth $0.995. A quick check of put–call parity suggests Synthetic put price = Call price + Present value of strike –Stock price = 5 + 0.995 × 55 − 55 = $4.725 And yet the price of the market-traded put was $9. Isn’t this an arbitrageur’s dream come true? Wouldn’t this mean that if you sell the market-traded put and buy the synthetic put (see expression [16.2]), you can immediately capture an arbitrage profit of 9 − 4.725 = $4.275 in one trade? Repeat a million times, and you become a millionaire four times over! Unfortunately, a market imperfection disrupted this trade. To create the arbitrage portfolio, an “arbitrageur” needs to sell the relatively overpriced market-traded put and “buy” the relatively underpriced synthetic put via buying the call, shorting the stock, and lending the present value of the strike. In this circumstance, however, the broker could not supply the Palm shares for short selling (by borrowing them from another investor). The scarcity of shares for short selling prevented the realization of the arbitrage profits and led to the divergence between theory and practice. The situation did not last long. The price anomaly disappeared within days; however, a lesson was learned. Paper profits can differ from actual realizable profits, and market imperfections do matter. 1

Palm IPO Soars, Then Retreats a Bit, Pushing Traders to Unwind Options in Parent 3Com,” W allStreetJournal, March 3, 2000.

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EXTENSION 16.2: Dividends and American Options

Put–Call Parity Adjusted for Dividends The parallels are eerie. One can easily modify the basic PCP by using the tools developed in Chapter 12 for adjusting the cost-of-carry model for dividends. The intuition remains the same as before. A dividend lowers the cost of a stock purchase, and replacing the stock price with the “stock price net of all dividends over the life of the derivative” modifies the formula. Assuming that a fixed-dollar dividend div is paid on a known future date t1 and B1 is today’s price of a zero coupon bond maturing on the ex-dividend date (recall Result 12.1a), put–call parity for European options (with a known dollar dividend) can be written as c = p + (S − B1 div) − BK

(1)

As before, this can be easily generalized to accommodate multiple dividends. For options on indexes, we can use the insights from Result 12.1b. If the underlying pays dividends at a continuous rate 𝛿, then put–call parity for European options (with a known dividend yield) can be written as c = p + e−𝛿T S − e−rT K

(2)

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A Put–Call Parity Inequality for American Options The basic PCP does not work for American options because the option may get exercised before the expiration date. However, a PCP inequality for American options does hold. For these inequalities, we add the assumption that interest rates are nonnegative, or equivalently, that a zero-coupon’s price is always less than or equal to 1.

EXT. 16.2 EX. 1: A Relation between American Calls and Puts

The Data ■

The stock price is S today. Assume that the stock pays no dividends over the life of the options.

■

American calls and puts worth cA and pA , respectively, trade on the stock. They have the same strike price K and expire at time T but may get exercised early at time t.

■

Interest is compounded continuously at the risk-free rate r percent per year. You can invest in zero-coupon bonds or a money market account (mma). B is today’s price and B(t) is the price at time t of zeros that mature at time T. An investment of $1 in a mma today grows to [1 + R(t)] on the intermediate (exercise) date t and to (1 + R) on the expiration date.

When the Portfolio Involves a Long Put (and Short Call) ■

Create a portfolio similar to the basic PCP: buy one put, write one call, buy one stock, and short bonds to borrow the present value of the strike. Today’s portfolio value is pA − cA + S − BK

(3)

MARKET IMPERFECTIONS

345

You decide when to exercise the put. Your counterparty decides when to exercise the American call. Thus two things can happen in the future: the holder of the American call exercises early, or she does not. Your response depends on Long’s decision on the call. ■

If the American call is not exercised early, then the American call becomes equivalent to the European call worth c. You can write cA = c. Now, if you don’t exercise your American put early, then it behaves like a European put, which means pA = p. With cA = c and pA = p, expression (3) becomes equal to 0 by PCP for European options. But you can exercise your American put before it expires. This flexibility makes the American put worth at least as much as the European put. Thus (3) becomes pA − cA + S − BK ≥ p − c + S − BK = 0

■

(4a)

If your counterparty exercises the American call early, then portfolio (3) has a payoff on the exercise date t (when you buy back the bonds by paying B[t] for each zero): pA (t) − [S (t) − K] + S (t) − B (t) K = pA (t) + [1 − B (t)] K ≥ 0

(4b)

This is nonnegative because a put cannot have a negative value and B(t) is worth less than or equal to a dollar. ■

Thus the portfolio (pA − cA + S − BK) has a value greater than or equal to zero. Combining (4a) and (4b) and moving cA to the right side, we get our first inequality

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pA + S − BK ≥ cA

(5)

When the Portfolio Involves a Short Put (and Long Call) ■

Create the portfolio: sell the put, buy the call, short the stock, and invest K dollars in a mma. Notice that this differs from the basic PCP portfolio because more dollars are kept in case the put is exercised early. This portfolio’s value today is − pA + cA − S + K (6) Here the buyer determines whether to exercise the American put. As before, we will act in response to Long’s decision.

■

If there is no early exercise, then the American put is the same as the European put, pA = p. If you don’t exercise the American call early, then it behaves like the European call, which means cA = c. With cA = c and pA = p, (6) becomes equal to KR on the maturity date (you can verify this in a payoff table). Because the last quantity is greater than zero, (6) cannot be less than zero today: − pA + cA − S + K ≥ 0

(7a)

Because you can exercise your American call early, this makes it even more valuable, which reinforces the relationship in (7a).

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If the buyer exercises early the American put, then portfolio (6) has a payoff on the exercise date t (when K has grown to K[1 + R(t)]): − [K − S (t)] + cA − S (t) + K [1 + R (t)] = KR (t) + cA ≥ 0

(7b)

because both the call and R(t) are nonnegative. ■

Thus the portfolio (− pA + cA − S + K) cannot have a negative value today. Combining (7a) and (7b) and rearranging terms, we get our second inequality: cA ≥ pA + S − K

■

(8)

For the PCP inequality for American options, combining (5) and (8), we get the relation between the stock, the bond, and the American options pA + S − K ≤ cA ≤ pA + S − BK

16.5

(9)

Options Price Restrictions

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This section explores various options price restrictions for several reasons. ■

They hone intuition and enhance understanding.

■

They generate arbitrage opportunities when the bounds are violated.

■

Just as you stretch before you sprint, a discussion of options price relations forms the stepping-stone to understanding early exercise of options, which is the final topic of this chapter.

To facilitate understanding, we establish the options price restrictions in several different ways: (1) a diagrammatic approach with profit diagrams; (2) an arbitrage table approach, where cash flows are recorded in a table and no arbitrage yields the result; and (3) a superglue argument, which is our name for comparing the original security with a “restricted” but otherwise identical security and then proving the result by arguing that “less cannot be worth more.” Why “superglue?” Read extension 16.3. We begin with the simple and then move to the complex. We use our standard notation and assumptions, except one: we allow the underlying stock to pay dividends. This is an important extension because most stocks pay dividends.

OPTIONS PRICE RESTRICTIONS

RESULT 16.2 American Options Are Worth More Than European Options 2a. 2b.

cA ≥ c ≥ 0 pA ≥ p ≥ 0

(16.5a) (16.5b)

This result holds because European and American options are identical, except for their exercise features. European options can only be exercised at expiration, whereas American options can be exercised anytime. As “more cannot be worth less,” an American option can never be priced less than an otherwise identical European option. This simple yet robust principle underlies the superglue argument, which is used later to prove some results (see Extension 16.2). Of course, an option’s price must be nonnegative because the holder can always discard the option without exercising it. The next two results restate the option price bounds established earlier in Chapter 5.

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RESULT 16.3 Call Price Boundaries 3a. An American call’s price is less than or equal to the stock price, that is, cA ≤ S

(16.6)

3b. When the stock price is zero, the call price is also zero.

Result 16.3 also holds for European calls because they are less valuable than American calls. Notice that an American call’s price cannot fall below the boundary condition stated earlier in Chapter 5. These are depicted in the first diagram in Figure 16.2, which reproduces Figure 5.3. Result 16.3b follows by setting S = 0 in Result 16.3a and recognizing that the call’s value must always be nonnegative.

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RESULT 16.4 Put Price Boundaries An American put’s price is less than or equal to the strike price K, and a European put’s price is less than or equal to the present value of the strike price: 4a. pA ≤ K (16.7a) 4b. p ≤ BK (16.7b) where B is today’s price of a zero-coupon bond that pays $1 at the option’s maturity.

A put option entitles the holder to receive the strike price, but he has to surrender the stock in return. Thus the put price cannot exceed the strike price, which forms an upper bound. Because a European put’s maximum payoff is the strike and only on the expiration date, today’s put price must be less than or equal to the present value of the strike price.

RESULT 16.5

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Call Price Bound before Expiration Before expiration, a European call’s price is greater than or equal to the larger of the stock price minus the present value of the strike or zero: c ≥ max (S − BK, 0)

(16.8)

We will prove this result with payoff diagrams, by an arbitrage table, and using the superglue argument (see Extension 16.3). The result also holds for American options, which are more valuable than otherwise identical European options. Figure 16.3 shows two payoff diagrams on the expiration date. The first diagram is for a stock purchase plan that consists of buying the stock and shorting K zerocoupon bonds. The second diagram, which is placed beneath the first, is for a long call. Visual inspection shows that the stock purchase plan pays off less than the long call. A consideration of today’s value of these two payoffs establishes that c ≥S − BK. Because the call’s value must also be nonnegative, we get expression (16.8). Our next example uses a variant of our data–model–arbitrage approach to establish the relation.

OPTIONS PRICE RESTRICTIONS

FIGURE 16.2: Price Bounds for American Calls and Puts Prices

Upper Bound Lower Bound Call prices in here

0

45° K

S, stock price

Prices Upper Bound

K Put prices in here

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0

K

Lower Bound S, stock price

EXAMPLE 16.3: Establishing Result 16.5 with an Arbitrage Table

The Data, Arbitrage, and Diagram ■

Consider the data from Example 16.2 with a slight modification. OPSY’s stock price S is $22.50, the six-month $20 strike European call price c is $2 and the put price p is $0.50, and the zero-coupon bond price B is $0.975310.

■

Does Result 16.5 hold? We have S − BK = 22.50 − 0.9753 × 20 = $2.9938. This is greater than the call price of $2, and the inequality in (16.8) is violated. Transferring the right side terms to the left side, we get the expression c − S + BK = 2 − 2.9938 = −$0.9938.

■

Buy low, sell high. Create an arbitrage portfolio by buying the underpriced call and selling the overpriced portfolio (S − BK) by shorting the stock and buying the bonds. Record these cash flows in Table 16.2. You immediately get $0.9938. At expiration, the portfolio has a 0 payoff when the call is in-the-money and a nonnegative payoff of [20 − S(T)] otherwise.

■

You can also see this arbitrage possibility by drawing the profit diagram for the portfolio in Table 16.2. Take $0.9938 and invest it in bonds to get $0.0190 after six months. The portfolio payoff after six months is [20 − S(T)] when the call is out-of-the-money, and 0 otherwise. Adding $1.02 (after rounding) to this shifts the whole curve up (see the diagram in Figure 16.4). This confirms the arbitrage opportunity.

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To establish the result, one simply replaces the numbers with algebraic symbols. Writing S = $22.50, K = $20, B = $0.975310, and c = $2, the last row in the second column of Table 16.2 shows that − c + S − BK = 0.9938. You can generalize the last row, as c < S − BK leads to a cash inflow today and a nonnegative payoff [K − S(T)] when S(T) ≤ K, and 0 otherwise. This is an unstable situation. To prevent arbitrage, we must have c ≥ S − BK (16.9) This is Result 16.5 when combined with the fact that c ≥0. Figure 16.4 also demonstrates an easy way to identify an arbitrage portfolio by drawing a profit diagram and checking to see whether you can make arbitrage profits.

FIGURE 16.3: The Call Price Must Exceed the Stock Price Minus the Present Value of the Strike Price (Result 16.5) Payoff

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Long stock

0 K = 20

S(T)

–20 Payoff Long call

0 K

S(T)

OPTIONS PRICE RESTRICTIONS

TABLE 16.2: Arbitrage Table Showing How a Violation of c ≥ S – BK (Result 16.5) Creates an Arbitrage Opportunity Portfolio

Today (Time 0) Cash Flow

S(T) ≤ 20

20 < S(T)

Buy call

−2

0

S(T) − 20

Short sell stock

22.5

−S(T)

−S(T)

−0.975310 × 20

20

20

0.9938

20 − S(T)

0

Buy bonds to lend the present value of strike (K = $20, B = $0.975310) Cash flows

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Expiration Date (Time T) Cash Flow

■

If the profit graph for a portfolio lies entirely above (or with some portions of it lying along) the horizontal axis, then you have created an arbitrage portfolio.

■

If the profit graph lies entirely below (or with some portions of it lying along) the horizontal axis, just reverse the trades to capture arbitrage profits.

■

A profit graph that crosses the axis leads to both trading profits and losses, which can never represent an arbitrage opportunity.

FIGURE 16.4: Profit Diagram Showing an Arbitrage Opportunity When the Call Price Is Less Than the Stock Price Minus the Present Value of the Strike Price (Violation of Result 16.5) Profit

21.02

1.02 0 K = 20

S(T)

351

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CHAPTER 16: OPTION RELATIONS

Result 16.5 has its counterpart for put options.

RESULT 16.6 Put Price Bound before Expiration 6a. Before expiration, a European put’s value must be greater than or equal to the larger of the present value of the strike price minus the stock price or zero: p ≥ max (BK − S, 0) (16.10a) 6b. Before expiration, an American put’s value must be greater than or equal to its intrinsic value: pA ≥ max (K − S, 0) (16.10b)

Result 16.6a follows easily using the superglue argument (see Extension 16.3). Result 16.6b follows from the boundary condition because you can always exercise an American put early and collect the strike price by surrendering the stock.

RESULT 16.7

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Relation between Options with Different Strike Prices 7a. The lower the strike price, the more valuable the European call: c (K2 ) ≤ c (K1 ) for K1 < K2

(16.11)

7b. The higher the strike price, the more valuable the European put: p (K1 ) ≤ p (K2 ) for K1 < K2

(16.12)

Both these results hold for American options as well.

Extension 16.3 proves several of these results using the superglue argument.

EXTENSION 16.3: The Superglue Argument Remember superglue, the unusually strong adhesive that advertisements claim can hang a car and lift two thousand pounds per square inch? Our use of superglue is different—we take the idea, but not the product! Consider two identical securities, which we name Sec. We attach a “restriction” to one of these securities, something

OPTIONS PRICE RESTRICTIONS

353

unfavorable or adverse that removes one of its provisions, and assume that it gets fastened with superglue (see Ext. 16.3 Fig. 1). As such, the restriction is permanent, and the security’s provisions are changed forever.

EXT. 16.3 FIG. 1: The Superglue Argument for Options SecR GLU

Sec E

Restriction

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Security

Security + Restriction

SecR is identical to Sec except for a restriction that is attached by superglue. Then, price of SecR < _ price of Sec.

The essence of the superglue argument is this: consider a security Sec and an otherwise identical security with the restriction attached, which we call SecR (short for SecurityRESTRICTED ). A long position in SecR cannot have a greater value than Sec because less cannot be worth more, that is, Price of SecR ≤ Price of Sec. This is a no-arbitrage condition. If it happens otherwise, then buy the security, add the restriction (with superglue at no cost), and sell it for more to make arbitrage profits. We can use this principle to derive all of our options price bounds and relations. We illustrate just a few of these below. Consider the proof of result 16.2 (directly below). Let Sec be an American option. Next, we attach a restriction to this option with superglue: “it cannot be exercised early.” Then, by no arbitrage, we know that the price of the restricted security—equivalent to a European call—must be less than the price of the unrestricted security—the American call. (Figure 16.5 illustrates this approach).

Result 16.2 American Options Are Worth More Than European Options Proof: Sec: American option. SecR : American option + Restriction “No early exercise allowed” Because SecR has become an otherwise identical European option, the superglue argument gives: European option price ≤ American option price.

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CHAPTER 16: OPTION RELATIONS

Result 16.3b cA ≤ S Proof: Sec: Stock SecR : Stock + Restriction “Pay an additional K dollars to maintain possession, otherwise throw away, and the decision must be made before time T.” Of course, with this restriction attached, the restricted stock is an American call. Hence American call price ≤ Stock price.

Result 16.5 c ≥ S – BK Proof: Sec: European call. SecR : European call + Restriction “Always exercise the call even if it is out-of-the money.” Today’s price of SecR is S − BK. The superglue argument gives the result. Result 16.2 tells us that this argument also holds for American calls.

Result 16.7a c(K1 ) ≥ c(K2 ) for K1 < K2

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Sec: European call with strike price K1 . SecR : European call with strike price K1 + Restriction “Must pay the extra amount (K2 − K1 ) in case of exercise.” The restricted security is equivalent to a European call with strike price K2 . The superglue argument gives the result. A similar argument holds for American calls.

EXAMPLE 16.4: Result 16.7b by the Modified Data–Model–Arbitrage Approach ■

Consider the COP data from chapter 15 with minor modifications. Puts trade on OPSY that mature in six months. The premiums are p(20) = $2 for strike price K1 = $20 and p(22.50) = $1.50 for K2 = $22.50. These numbers violate inequality (16.12). We use the arbitrage opportunity in the data to justify our model, which is relation (16.12).

When Both Puts Are European ■

As shown in Table 16.3, we create the arbitrage portfolio by selling the overpriced OPSY 20 put and buying the underpriced OPSY 22.50 put. This gives a cash flow of + 2 − 1.50 = $0.50 today. The puts’ payoffs on the expiration date depend on the stock price. - If S(T) is less than or equal to $20, the short put has a payoff of −[20 − S(T)] and the long put has a payoff of [22.50 − S(T)]. This gives a payoff of $2.50. - For 20 < S(T) ≤ 22.50, the OPSY 20 put has a 0 payoff while the long OPSY 22.50 put has a payoff of [22.50 − S(T)]. For this range for S(T), the payoff is nonnegative. - When S(T) is greater than $22.50, both options are out-of-the-money and have 0 payoffs.

■

We showed that p (20) = $2 > p (22.50) = $1.50

355

OPTIONS PRICE RESTRICTIONS

leads to an arbitrage opportunity. The no-arbitrage requirement dictates that p(20) ≤ p(22.50). ■

Replace numbers with symbols and you get the result p (K1 ) ≤ p (K2 ) for K1 < K2

When Both Puts Are American ■

Suppose the mispricing holds and the prices reflect those for American puts instead. As with European puts, create an arbitrage portfolio by selling the OPSY 20 American put and buying the OPSY 22.50 American put. This gives $0.50 today. In the future, if the short put is not exercised against you, hold on to the long put. The payoff on the expiration date is given in Table 16.3. By contrast, if the OPSY 20 put is exercised against you, exercise the OPSY 22.50 put to eliminate the stock position, and you are left with $2.50 on the exercise date. This trade is an arbitrage opportunity. Hence, the only American put prices consistent with no arbitrage must satisfy result 16.7b.

TABLE 16.3: Arbitrage Table for Result 16.7b: The Higher the Strike Price, the More Valuable Is the Put Expiration Date (Time T) Cash Flow

Portfolio

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Sell 20 put Buy 22.50 put Cash flows

S (T) ≤ 20

20 2.50 because B < 1. Given these thought experiments, we now better understand Result 16.10a. It says that you can make more profits by selling the OPSY call rather than by exercising it. Thus, if no dividends are paid to the stockholder over the option’s life, then the call option is always worth more alive than dead. Do not exercise such a call: sell it. Sell it! SELL IT!! Puts, though, have a different story. Unlike calls that have an unbounded profit potential, put payoffs are capped by the strike price, the maximum payoff possible. This leads to Result 16.10b. To understand why Result 16.10b is true, consider the following hypothetical example. Consider a six-month original maturity put on OPSY that began two months back. If OPSY’s stock price is S(t) = $0.0001 today, an immediate exercise gives K − S(t) = (20 − 0.0001) = $19.9999. If the interest rate r is 6 percent per year, investing the proceeds in a money market account over the next day yields [K − S (t)] erT = [19.9999] e0.06 × (1/365) = $20.0032

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Waiting one day and then deciding to exercise yields at most $20, and if one waits, the stock price may rise and even less of a payoff may occur. So early exercise is optimal today. Of course, this is a pathological example selected to make our point, but it does prove that early exercise is sometimes optimal if the stock price is low enough and interest rates are high enough. This pathological example also illustrates another observation. It is easy to believe that as the stock price rises, there is some stock price S* where you are just indifferent between exercising today or not. The S* just balances the interest earned on the proceeds of early exercise with the chance of the stock price declining more (see Figure 16.6). Such an S* does in fact exist. Unfortunately, to prove this observation and to determine the S* precisely, we need a model for the stock price evolution. This analysis will be pursued in chapter 18. For now, we can only prove Result 16.10b.

Dividends and Early Exercise Dividends change the optimal exercise decision for American options. Recall that when dividends are paid, the stock price falls. We characterized this dividend-induced price drop as Result 3.1:3 SCDIV = SXDIV + div (3.1) where SCDIV is stock price cum-dividend, SXDIV is stock price ex-dividend, and div is the amount of dividend. A natural place to investigate early exercise is around the time a stock goes ex-dividend. This leads to our next result.

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FIGURE 16.6: Illustration of Result 16.10b Prices

K = 20 Stock price S* 0 t T Early exercise

Time

At S*, one is indifferent between exercising or not. Lower stock prices give higher proceeds from early exercise.

3

Exceptions to this statement can happen. We need some technical assumptions about the process underlying stock price changes. The statement in (3.1) holds, for example, if the stock price process follows a lognormal distribution, which underlies the Black–Scholes–Merton model. It does not hold, for example, if the stock price process allows discrete jumps and the jump happens at the same time as the stock goes ex-dividend; see Heath and Jarrow (1988) for a formal discussion.

EARLY EXERCISE OF AMERICAN OPTIONS

RESULT 16.11 Optimal Times for Exercising American Calls The only times when it may be optimal to exercise early is just before a stock goes ex-dividend.

The next example establishes this result with an application of Result 16.5.

EXAMPLE 16.6: Establishing Result 16.11 ■

Consider a six-month OPSY 20 call. Let the next dividend div1 be paid at time t1 , which is two months from today (time 0). Should you exercise this call at a time t, anytime over the next two months?

■

Consider the two possible strategies: early exercise at time t or wait and exercise at time t1 . If you exercise at time t, the value of your position is S(t) − 20, but if you exercise an instant before OPSY goes ex-dividend at time t1 , even if the option is out-of-the-money, then you get SCDIV − 20. The present value at time t of waiting until time t1 to exercise is

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S (t) − 20B where B is the time t price of a zero-coupon bond that pays $1 at time t1 . ■

Because B is less than 1, the second strategy gives you more value than the first. Thus exercising the American call just before the stock goes ex-dividend is better than exercising it anytime between now and the first dividend date.

■

This result follows because exercising earlier than these ex-dividend dates leads to an early surrender of the strike price—an action that loses interest but does not generate any additional benefit. Hence, if the call is exercised early to obtain the dividend on the stock, it will be exercised just an instant before the stock goes ex-dividend. This is depicted in Figure 16.7. See Jarrow and Turnbull (2000) and other finance textbooks for a discussion of how big the dividend needs to be to trigger early exercise.

And what about developing early exercise results for puts? Puts can get exercised early even without dividends. As dividends would cause the stock price to drop further, their presence is likely to delay early exercise. To get more precise insight, we need to model the stock price evolution. Again, we address this in later chapters.

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FIGURE 16.7: Timeline for Exercise of American Options If exercise call, exercise at these points

Option starts

Ex-div (div1)

Ex-div (div2)

Expiration

Exercise put anywhere If exercising an American call early, exercise just before the stock goes ex-dividend. An American put may be exercised anytime (but dividends tend to delay exercise).

16.7

Summary

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1. Put–call parity for European options (see Result 16.1) states that the call price plus the present value of strike equals the stock price plus the put price, or c + BK = S + p, where B is today’s zero-coupon bond price paying a dollar on the options’ common maturity date and K is the common strike price. 2. PCP has a number of applications and uses: - PCP can be used to create synthetically an option, a stock, or a bond. - PCP can be easily modified to adjust for known dividends. 3. Option prices satisfy restrictions that can be derived using the no-arbitrage principle. These theorems hone intuition and enhance understanding, lead to arbitrage profits in case of violations, and form a stepping-stone to understanding the early exercise of options. 4. Options price restrictions can be established in several different ways: (1) a diagrammatic approach with profit diagrams, (2) an arbitrage table approach, where cash flows are recorded in a table and no arbitrage yields the result, and (3) a superglue approach, which is our name for comparing the original security with a “restricted” but otherwise identical security and then proving the result by arguing that “less cannot be worth more.” 5. Some options price properties for stocks that may pay dividends are as follows: - Result 16.2—American options are worth more than European options: cA ≥ c and pA ≥ p. - Result 16.3—Call price boundaries 3a An American call’s price is less than or equal to the stock price: cA ≤ S. 3b When the stock price is zero, the call price is also zero.

SUMMARY

- Result 16.4—Put price boundaries 4a An American put’s price is less than or equal to the strike price K: pA ≤ K. 4b A European put’s price is less than or equal to the present value of the strike price: p ≤ BK. - Result 16.5—Before expiration, a European call price must exceed the stock price minus the present value of the strike price: c ≥ S − BK. - Result 16.6a—Before expiration, a European put must be greater than or equal to the present value of the strike price minus the stock price: p ≥ BK − S. - Result 16.6b—Before expiration, the value of an American put must be greater than or equal to its intrinsic value: pA ≥ K − S. - Result 16.7a—The lower the strike price, the more valuable is the European call: c(K1 ) ≥ c(K2 ) for K1 < K2 . - Result 16.7b—The higher the strike price, the more valuable is the European put: p(K1 ) ≤ p(K2 ) for K1 < K2 . These results hold for American options as well. - Result 16.8—Option prices and time to maturity: the longer the time until expiration, the greater the price of an American call or a put.

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8a cA (T1 ) ≤ cA (T2 ) for T1 < T2 . 8b pA (T1 ) ≤ pA (T2 ) for T1 < T2 . However, these results need not hold for European calls or European puts. 6. Early-exercise considerations begin with the following: Result 16.9a—Exercise an American call the first time its price is equal to the stock price minus the strike price: cA = S − K. Result 16.9b—Exercise an American put the first time its price is equal to the strike price minus the stock price: pA = K − S. 7. Dividends play a critical role in early-exercise decisions: - Assume the stock pays no dividends over the option’s life: Result 16.10a—The American call becomes identical to a European call. It should never be exercised early. Result 16.10b—The American put should be exercised if the stock price is small enough relative to the strike price and the time to maturity. - Absent market frictions, dividends cause the stock price to drop by the amount of the dividend (Result 3.1). This is used to establish the following: Result 16.11—The only times when it may be optimal to exercise an American call early is just before a stock goes ex-dividend. Puts can get exercised early even without dividends, but dividends are likely to delay early exercise.

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16.8

Cases

Boston Properties (A and B) (Harvard Business School Cases 211018 and 211041-

PDF-ENG). The case introduces options pricing, payoff diagrams, and the law of one price and explains arbitrage as well as no-arbitrage bounds. Smith, Barney, Harris Upham and Co. Inc. (Darden School of Business Case

UV0074-PDF-ENG, Harvard Business Publishing). The case approaches put–call parity from the trader’s perspective and examines the practical aspects of doing arbitrage in the options markets. Sleepless in L. A. (Richard Ivey School of Business Foundation Case 905N11-

PDF-ENG, Harvard Business Publishing). The case discusses the Black–Scholes– Merton model for options pricing, the concept of implied volatility, and put–call parity. It also shows how options pricing can be used to value corporate liabilities of a financially distressed company.

16.9

Questions and Problems

The next two questions are based on the following data for European options: Call price = $5, risk-free continuously compounded interest rate r = 5 percent per year, stock price S = $55, strike price K = $55, time to maturity T = 1 month. 16.1. If the put price p = $9, show how to capture arbitrage profits in this market. 16.2. Suppose short selling of stocks is not allowed in this market. Can you still

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make arbitrage profits? Explain your answer. 16.3. Do the following data satisfy put–call parity for European options? If they

don’t, show how you can create a portfolio to generate arbitrage profits. Call price = $6, put price = $3, stock price S = $102, strike price K = $100, time to maturity T = 3 months, and risk-free continuously compounded interest rate r = 5 percent per year. 16.4. How can you adjust put–call parity for known dividends on a single known

date? 16.5. Prove put–call parity for European options in the case of a single known

dividend: c + PV (Div) + Ke−rT = p + S where S is stock price, K is strike price, T is maturity date for the option, r is risk-free interest rate, c is European call price, p is European put price, and PV(Div) is the present value of dividends. 16.6. Using put–call parity, given c = $2, PV(Div) = $1, p = $1, S = K = $100,

r = 0.05 per year, and T = 0.25 years, can you make arbitrage profits? Explain. 16.7. Explain the relation between a put option (with strike price K and maturity

date T years from today) and a T-period insurance policy that insures the stock for K dollars.

QUESTIONS AND PROBLEMS

16.8. Different countries, different customs, different market practices! Suppose you

go to a country where traders with inside information can easily trade. There, you have inside information that the stock price is going to increase for sure (no chance that it will decline), and it is legal to trade on this information. What is the best strategy to use? 16.9. Does put–call parity always hold in financial markets? If not, give a few reasons

why it may not hold. 16.10. On the day options on Palm began trading, the share prices grossly violated

the put–call parity. Describe and explain why this happened. 16.11. Can you link Russell Sage’s actions with a distinguished economist’s view on

what drives financial innovation? 16.12. Is it true that the lower the exercise price, the more valuable the call? Explain

your answer. 16.13. Is a European put on the same stock with the same maturity worth more or

less if the strike price increases? Explain your answer. 16.14. Is it true that the more the time until expiration, the less valuable an American

put? Explain your answer. 16.15. Can you make arbitrage profits from the following European call prices?

If so, give two such examples of arbitrage, neatly showing the portfolio construction as well as the various cash flows. The stock price is $40.

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Strike Price

Expiration Month April

July

September

35

1

6

3

40

2

5

6

16.16. The following prices are given for American put options on a stock whose

current price is $100: Strike Price

Expiration Month March

June

95

1

11

100

7

5

Construct three portfolios for making arbitrage profits, showing the cash flows from each portfolio.

365

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16.17. The following prices are given for American call options on a stock whose

current price is $100: Strike Price

Expiration Month March

June

95

11

10

100

8

8

105

2

5

Construct three portfolios for making arbitrage profits, showing the cash flows coming from each portfolio. 16.18. If an American call is written on a stock that never pays a dividend, would

you ever exercise the call option early? Explain your answer. 16.19. If an American put is written on a stock that never pays a dividend, would

you ever exercise the put option early? Explain your answer. 16.20. a. Is there a simple rule of thumb that you can use to know when to exercise

an American call early? If yes, explain the rule. b. Is there a simple rule of thumb that you can use to know when to exercise

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an American put early? If yes, explain the rule.

17

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Single-Period Binomial Model 17.1 Introduction

Arbitrage-Free Trees

17.2 Applications and Uses of the Binomial Model

Stock Prices and Martingales

17.3 A Brief History of Options Pricing Models Option Pricing Pre-1973 Options Pricing, 1973 and After

17.4 An Example 17.5 The Assumptions 17.6 The Single-Period Model The No-Arbitrage Principle Building Binomial Trees

The Pricing Model The Hedge Ratio Risk-Neutral Valuation Actual versus Pseudo-probabilities

17.7 Summary 17.8 Appendix Proving the No-Arbitrage Argument The Probabilities and Risk Premium

17.9 Cases 17.10 Questions and Problems

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CHAPTER 17: SINGLE-PERIOD BINOMIAL MODEL

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17.1

Introduction

“Once in Hawaii I was taken to see a Buddhist temple,” wrote physicist Richard Feynman in The Meaning of It All: Thoughts of a Citizen Scientist. “In the temple a man said, . . . ‘To every man is given the key to the gates of heaven. The same key opens the gates of hell.’” Derivatives may be viewed as such a key. When the financial crisis of 2007 hit many countries, including the United States, numerous commentators, and even the lay public, blamed it on derivatives. Indeed, credit rating agencies failed to correctly rate complicated derivatives, and the losses on these derivatives brought down many financial institutions. Conversely, the growth of the derivatives market has been praised as improving economic welfare by shifting risks from those who fear it to those who profit from it. And derivatives help solve financial problems. They were even used by governments during the crisis to help resolve it. For example, the US Treasury started the Troubled Asset Relief Program during the financial crisis to commit up to “$700 billion to rescue the financial system,” notes the July (2009) Oversight Report of the Congressional Oversight Panel, “TARP Repayments, Including the Repurchase of Stock Warrants.” TARP purchased stock in the stressed banks with “troubled assets,” but in addition, they received ten-year warrants. Warrants are call options—derivatives! After the banks recovered, the US government sold these warrants back to the banks or to others at a “fair price,” determined with the help of the Black–Scholes–Merton (BSM) and binomial models. In doing so, they profited substantially. Note the use of the binomial model! This chapter develops the single-period binomial model with an eye toward understanding the major tenets of options pricing. We provide an illustrative example, give the model’s intuition, and state the necessary assumptions. To value the option, we construct a synthetic option with identical payoffs using the stock and a money market account. This takes us to martingale pricing and risk-neutral valuation, which lies at the heart of modern theory and which enables us to understand the more complex option pricing models (OPMs) that follow. Chapter 18 extends this chapter to a multiperiod setting and makes the binomial model practical and useful. The subsequent two chapters adopt a similar pattern: chapter 19 introduces the BSM model, and chapter 20 discusses the model’s practical use.

17.2

Applications and Uses of the Binomial Model

Options pricing bewildered and baffled academics until Fischer Black, Myron Scholes, and Robert Merton developed an analytic pricing model for European options in 1973. The mathematics underlying the model’s derivation is difficult. To more simply derive the model, as William Sharpe noted, one can use a binomial model and then take limits. Although the binomial model had been used as a teaching tool at the Massachusetts Institute of Technology (MIT) and other places before its

A BRIEF HISTORY OF OPTIONS PRICING MODELS

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publication, it was first printed in Sharpe’s classic textbook Investments. Subsequently, Cox, Ross, Rubinstein, Rendleman, Bartter, Jarrow, and Rudd developed popular versions of the binomial model. The model gets its name binomial (bi means “two”) from the assumed stock price evolution. From any point in time onward, over the next time step, the stock price can take only one of two possible values. This raises two immediate questions: (1) why do we need to make an assumption about stock price movements, and (2) is it reasonable to assume that stock prices can only take two values? The last chapter explicitly answered the first question. There we pushed as far as we could without imposing a model for the stock price evolution. Although we got many results, we could not obtain a pricing model. One needs to assume a model for the stock price evolution like the binomial (which we do in this and the next chapter) or a lognormal (chapter 19) to develop an exact OPM. As for the second question, do stocks really only take one of two values at each time step? No. But we need to start somewhere. We start with the simplest model to facilitate understanding and then add the necessary layers of complexity. The simple binomial OPM possesses some desirable features: ■

It approximates the BSM model. The binomial OPM can be repeated at each time step in a multiperiod tree. When the parameters are properly chosen (see Extension 18.1) and the model is run over many small time intervals, the multiperiod binomial OPM prices closely approximate BSM model prices.

■

It illustrates martingale pricing, the central idea of derivatives pricing. The binomial approach utilizes “martingale pricing,” a powerful technique for valuation. Once you understand this approach, you will find it easier to understand more complex OPMs.

■

It provides a useful numerical approximation technique. The binomial model is a versatile tool that gives numerical values to approximate solutions to exact pricing models (like the BSM model) or provides answers when no explicit analytical solutions exist (as in the case of an American put). Later chapters show how to use the binomial model to price interest rate derivatives. Even proprietary trading desks of Wall Street firms sometimes use sophisticated versions of the binomial pricing model.

17.3

A Brief History of Options Pricing Models

The history of OPMs has two phases: before and after 1973.

Option Pricing Pre-1973 It all started more than a century ago, when a young French mathematician, Louis Bachelier (1900), wrote a dissertation titled “The Theory of Speculation” that

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developed a prototype OPM. Despite some shortcomings, it was the forerunner of the BSM model. Bachelier was the first to model stock prices as a random walk. But fate was unkind, and he did not get the recognition he deserved during his lifetime. Within a decade after Bachelier’s death, his work resurfaced. It attracted the attention of Professor Paul Samuelson, the first American to win the Nobel Prize in Economics. During the 1960s, Samuelson and others worked on improving Bachelier’s model (see Table 17.1 for a summary of landmark achievements in this field).

TABLE 17.1: Some Milestones in the History of Option Pricing Models

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Year

Development

1900

Bachelier developed the first option pricing model (OPM).

1960s

Samuelson and other scholars improved Bachelier’s model.

1973

The Black–Scholes and Merton models were published.

1970s, 1995

Merton (1974, 1977) and Jarrow and Turnbull (1995) models for pricing derivatives with credit risk were introduced.

1978–82

The binomial model was presented in Sharpe (1978), Cox et al. (1979), Rendleman and Bartter (1979), and Jarrow and Rudd (1982).

1979 and 1981

Harrison and Kreps (1979) and Harrison and Pliska (1981) developed the martingale pricing methodology.

1987 and 1992

The Heath–Jarrow–Morton model (introduced in 1987, published in 1992), an interest rate OPM, was introduced.

1997

Merton and Scholes, the surviving co-originators of the BSM model, won the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.

Options Pricing, 1973 and After During the late 1960s, two young PhDs, Fischer Black and Myron Scholes, teamed up to work on options pricing. Around the same time, Robert Merton, a PhD student of Paul Samuelson, started working on the same problem. Combining their insights, 1973 saw the publication of the Black–Scholes–Merton model (Chapter 19 provides a more detailed history), which forms the foundation for today’s OPMs. Their work stands as a towering achievement not only in finance but also in the field of economics. After the watershed year of 1973, a race began to build better pricing models. Numerous OPMs resulted. These models priced different kinds of derivatives and relaxed the BSM model’s assumptions. The race continues to this day with unabated fury.

A BRIEF HISTORY OF OPTIONS PRICING MODELS

Why develop better OPMs? There are many reasons. The opening of options exchanges and the expansion of over-the-counter options trading created a need for better hedging. Brokers and dealers need OPMs to make effective marketsmarkets. The OPM extensions can be classified under five headings: 1. Improving the BSM model and using it for different underlyings. Soon after the publication of the BSM came a slew of models that improved the original. They included Black’s model to price commodity contracts. The stock price evolution was generalized, and market imperfections like short selling restrictions and transaction costs were included as well. 2. Including a term structure of random interest rates. A major drawback of the BSM model is its constant interest rate assumption. Early attempts at pricing interest rate options required estimating risk premia (e.g., Vasicek 1977; Brennan and Schwartz 1979) that made the models difficult to use. The removal of the need to estimate interest rate risk premia came in 1992 when the Heath–Jarrow–Morton (HJM) model was published. All arbitrage-free interest rate OPMs are special cases of the HJM model.1

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3. A rigorous mathematical foundation. Harrison, Kreps, Pliska, and others generalized the mathematics behind the BSM model. They developed martingale pricing, which soon became the most widely used technique for pricing derivatives. 4. Pricing derivatives with credit risk. Merton (1974, 1977) pioneered the pricing of derivatives with credit risk with the structural approach. Jarrow and Turnbull (1995) developed an alternative method that relaxes some restrictive assumptions inherent in Merton’s formulation. The Jarrow–Turnbull approach is called the reduced-form credit risk model. See chapter 26 for additional discussion. 5. Numerical methods. With more complex pricing models came the need to compute their values quickly and efficiently on computers. The binomial model was one of the first successful numerical methods (see Sharpe 1978; Cox et al. 1979; Rendleman and Bartter 1979; Jarrow and Rudd 1982). Others include numerical methods for solving partial differential equations and Monte Carlo simulation techniques. To understand options pricing theory, we begin with an example that illustrates the key ideas.

1

Baxter and Rennie (1996) state, “In the interest-rate setting, Heath–Jarrow–Morton is as seminal as Black–Scholes. By focusing on forward rates and especially by giving a careful stochastic treatment, they produced the most general (finite) Brownian interest-rate model possible. Other models may claim differently, but they are just HJM with different notations.”

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17.4

An Example

This section illustrates the binomial options pricing approach through an example (see Example 17.1).

EXAMPLE 17.1: European Call Pricing ■

Your Beloved Machine Inc.’s (YBM) stock price S is $100 today (time 0). After one year (time T), the stock price S(1) can either go up to $120.00 or down to $90.25. Let the actual probability of the stock going up q = 3/4; then that of YBM going down is (1 – q) = 1/4.

■

Consider a European call option with strike price K = $110 and maturity date time 1. The call’s payoff at time 1 is max[S(1) – 100, 0], that is, it is 10 = max(120 – 110, 0) if the stock goes up and 0 = max[90.25 – 110, 0] if the stock goes down.

■

Let a money market account (mma) cost $1 per unit at time 0. It earns a continuously compounded interest rate r = 0.05 per year. A dollar invested in the mma grows to $1.0513 (see Figure 17.1).

Synthetic call construction ■

Form a portfolio V with m shares of the stock and b units of the mma. Its value today is V (0) = m100 + b

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■

(17.1)

We want to choose (m, b) such that the portfolio’s payoffs at time 1, V (1), equal those of the traded call on the expiration date. This holds if the following two equations are satisfied: m120.00 + b1.0513 = 10

(17.2)

m90.25 + b1.0513 = 0

(17.3)

■

Solving these equations gives m = 0.3361 and b = –28.85, where a positive sign indicates a long and a negative sign a short. We call this portfolio the synthetic call.

■

Plugging the solutions into expression (17.1) gives V (0) ≡ m100 + b = $4.76. This is the cost of constructing the synthetic call at time 0. To avoid arbitrage, this must be today’s price for the traded call, c.

■

It is interesting to note that the option’s value was determined without using the actual up probability q = 3/4. This is an important observation. We will explain the reasons for this shortly.

■

Note that expression (17.1) also has a hedging interpretation. If you buy V (0) (a call option) and want to hedge the stock price risk, then sell m shares of the stock. This creates the hedged call portfolio V (0) – m100 = b, which is riskless because it’s equivalent to a position in the mma. We will also explain this hedging interpretation more completely later in this chapter.

Arbitrage Profits ■

What happens when the traded call’s price differs from the price of the synthetic call? Seize the opportunity and make arbitrage profits by buying low, selling high, or as the British would say, buying cheap, selling dear.

AN EXAMPLE

■

Suppose an errant trader quotes $7 for the traded call. As the traded call is more expensive, sell it and buy the synthetic call. Let’s do the accounting to see if it all works out. - Sell the overpriced traded call for $7 and purchase the underpriced synthetic call by buying 0.3361 shares for $33.61 and borrow $28.85 by shorting the mma at a net cost of $4.76. - Today’s cash flow is + 7 – 4.76 = $2.24. This is the immediate profit. - You have no liabilities in the future. If the stock rises at maturity, you owe $10 as the writer of the traded call. This is offset by the $10 that you get as the buyer of the synthetic call. Both calls expire worthless if the stock falls instead.

■

If a trader quotes $3 for the call, simply reverse your strategy: buy the traded call and simultaneously sell the more expensive synthetic call by short selling 0.3361 shares for $33.61 and lending $28.85 by buying the mma. You immediately make $1.76, and there is no future net exposure as the assets and liabilities perfectly match.

FIGURE 17.1: Single Period Binomial Example and Model Numerical Example Stock

Call

Today

After 1 year 3/4

Option starts

120.00

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100

Option matures

3/4

10

1/4

0

c

1/4 90.25 Call: strike price K = 110, maturity time 1 One-Period Model Stock

Call

Now

Maturity q

Time 0

US

Time 1 q

cU

(1 – q)

cD

c

S DS (1 – q) where U > D and 1 > q > 0

Call: strike price K, maturity time 1, payoff c(1) = max[S(1) – K, 0] Money Market Account (mma) 1 1.0513 1 + R er

t

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Put Pricing ■

Consider a European put with a strike price of K = $110 and a maturity of one year. As with the call, set up a portfolio to match the put’s time 1 value of $0 = max(110 – 120, 0) in the up state and $19.75 = max(110 – 90.25, 0) in the down state. This yields two equations in two unknowns (m, b): m120.00 + b1.0513 = 0 m90.25 + b1.0513 = 19.75

■

Solve the two equations to get the stock shares m = – 0.6639 and the number of mmas needed b = 75.78.

■

As the stock and put prices move in opposite directions, it makes intuitive sense that the replicating portfolio combines a short position in the stock and a long position in the mma.

■

The European put option’s price p is equal to the cost of construction at time 0, that is, V (0) ≡ m100 + b = − 0.6639 × 100 + 75.78 = $9.39

Put–Call Parity ■

Alternatively, you can derive the put’s value using put–call parity for European options:

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Put price = Call price + Present value of strike − Stock price = 4.76 + 110 × (1/1.0513) − 100 = $9.39 Any other put price will open up arbitrage opportunities.

17.5

The Assumptions

Assumptions are the key to understanding and using any model. Recall our discussion of the cost-of-carry model and its assumptions from Chapter 11. We need those same assumptions (A1 to A5) plus two more (A6 and A7). One is needed to describe the behavior of interest rates, and the other is needed to characterize the binomial stock price evolution. For emphasis as well as a better understanding of the model, we repeat our discussion of assumptions A1–A5. A1. No market frictions. We assume that trading involves no market frictions such as transaction costs (brokerage commissions, bid/ask spreads), margin requirements, short sale restrictions, and taxes (which may be levied on different securities at different rates). Moreover, assets are perfectly divisible. This helps us to create a benchmark model to which one can later add frictions. In addition, it’s a reasonable approximation for many traders such as institutions.

THE ASSUMPTIONS

Moreover, political changes, regulatory changes, and the information technology revolution have substantially reduced market frictions.

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A2. No credit risk. We assume that there is no credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty will fail to perform on an obligation. The absence of credit risk is a reasonable assumption for exchange-traded derivatives. Over-the-counter derivatives, however, may possess substantial credit risks, unless there are adequate collateral provisions. A3. Competitive and well-functioning markets. In a competitive market, traders’ purchases or sales have no impact on the market price; consequently, traders act as price takers. In a well-functioning market, price bubbles are absent. The competitive markets assumption is the workhorse of modern economics. Spot, forward, and futures markets for many agricultural commodities and precious metals behave like competitive markets. Pricing securities when counterparties have market power and traders can manipulate prices is a difficult exercise. One must take into account bargaining and strategic interactions, which are outside the scope of this book. A price bubble happens when an asset’s market price deviates from its intrinsic or fundamental value. The fundamental or intrinsic value is the price paid if, after purchase, you have to hold the asset forever. A difference between the market price and fundamental value only occurs if one believes that selling can generate a higher value than holding it forever. The difference is the price bubble. Recent research shows that bubbles can invalidate the methodology we employ for pricing options. Fortunately, however, one can show that there are no price bubbles possible in a discrete time binomial model (see Jarrow et al. 2010). A4. No intermediate cash flows. Many assets reward their holders with income or cash flows such as dividends for stocks and coupon income for bonds. For now, we assume that the underlying asset has no cash flows over the option’s life. The next chapter will show how to relax this assumption by introducing dividends. A5. No arbitrage opportunities. This is self-explanatory. These five assumptions are a repetition of the assumptions used in the cost-ofcarry model of chapter 11. Next we introduce two additional assumptions needed for deriving the binomial OPM. A6. No interest rate uncertainty. We assume that interest rates are constant across time. This assumption only works for short-lived options whose underlyings are not interest rates. It does not work well for long-dated options whose underlying assets’ prices are correlated with interest rate changes, like long-term options on foreign currencies. But long-dated options and interest rate derivatives are important in today’s markets, and their pricing is a prime challenge for researchers and practitioners. We relax this assumption in part IV of the book. Recall that we did not require this assumption for our cash-and-carry models. It wasn’t necessary because forwards are essentially valued in a single-period setting. However, we will soon be introducing multiperiod and continuous time models in

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which bond investments have reinvestment risks. To avoid this problem, we assume a constant interest rate. Assumption A6 will be maintained for all OPMs in part III of the book. A7. Binomial process. We assume that the stock trades in discrete time and that its price evolves according to a binomial process. For a current stock price, next period’s stock price will either get multiplied by an up factor U > 1 and take a higher value U × Stock price or it will be multiplied by a down factor D and take a lower value D × Stock price. We let q > 0 denote the actual probability of going up. We also assume U > D. Otherwise, we have mislabeled up and down. Assumption A7 will be with us in this and the next chapter. It will eventually be replaced by the assumption of “continuous trading” and the evolution of the stock price according to a “lognormal process” in the context of the BSM model.

17.6

The Single-Period Model

To develop the model, we proceed through a sequence of steps. First, we explain the no-arbitrage principle used in the example. Second, we show how to build a binomial tree that is arbitrage-free. Next comes the martingale representation of stock prices, which helps us to develop fancy entities called pseudo-probabilities. Then we obtain the call’s value by building on Example 17.1. Finally, we introduce an alternative perspective on options pricing called risk-neutral valuation. The next chapter will extend this to develop a multiperiod binomial OPM that approximates the BSM model.

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The No-Arbitrage Principle Let’s summarize the no-arbitrage principle used in the previous example to find the option’s value. ■

We considered a market in which both a stock and an option traded.

■

The stock took one of two possible values at the maturity date, and so did the option because the option’s payoff is completely determined by the stock’s ending value and the strike price.

■

We created a portfolio of the stock and mma to match the option’s payoffs in each of these time 1 states. This replicating portfolio is called the synthetic option.

■

By the law of one price—the no-arbitrage principle—the cost of constructing the synthetic option must equal the price of the traded option.

■

Why should they be the same? If the traded option and the synthetic option differ in price (see Figure 17.2), then you have two distinct ways of getting the same cash flows. Shrewd traders will make arbitrage profits by buying the cheaper option and selling the dearer option until their prices converge.

■

This pricing methodology has an interesting by-product. A portfolio that holds one of these options long and the other short should be entirely riskless. This kills two birds with one stone: pricing and hedging problems are solved in a single stroke!

THE SINGLE-PERIOD MODEL

FIGURE 17.2: The No Arbitrage Principle Portfolio A (Market traded option)

Portfolio value in up state

Must be equal without arbitrage Portfolio B (Synthetic option)

Portfolio value in down state

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Building Binomial Trees We need some notation to build a binomial tree. Today at time 0, the stock price is S(0) or S. At time 1, it can move up to S(1)UP or slide down to S(1)DOWN . Let’s write S(1)UP = SU and S(1)DOWN = SD, where the positive numbers U > D are called the up and down factors, respectively (Figure 17.3 shows how a stock price path looks in a single-period tree). This multiplicative binomial model has the advantage of ruling out negative stock prices. Moreover, in the multiperiod setting, it makes life easier by insuring that the tree’s branches recombine. To reduce clutter, we drop the dollar sign from the tree. The other traded security is a mma. It earns the risk-free rate in each period. A dollar gives (1 + R) = exp(rΔt) ≡ erΔt at the end of each period, where r is the continuously compounded yearly interest rate and Δt is the length of the time

FIGURE 17.3: Stock Price Evolution in a Single Period Setting Stock price Realized stock price path

US S DS

Possible path

0 Today

Time 1

Time

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CHAPTER 17: SINGLE-PERIOD BINOMIAL MODEL

period. In Figure 17.1, a single arrow is used to show the mma’s value because it earns a fixed return. Consider a European call option with strike price K and maturity time 1. Let c(0) or c denote its time 0 value. Given the call’s boundary condition, the call’s values at date 1 are given by max[0, S(1) – K]. We label this as c(1)UP ≡ cU when the stock price goes up and c(1)DOWN ≡ cD when it goes down (we drop some letters to reduce clutter). Figure 17.1 shows this model under the binomial tree example. Put values pU and pD could have been similarly introduced.

Arbitrage-Free Trees

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“House built on a weak foundation, will not stand oh no,” sang “King of Calypso” Harry Belafonte, who popularized Caribbean music. The same can happen with a model. Because we price options with a binomial tree using the no-arbitrage principle, the tree must be arbitrage-free for logical consistency. Otherwise, the whole approach collapses, like a house without a strong foundation. A simple condition Up factor > Dollar return > Down factor (U > [1 + R] > D) is both necessary and sufficient to rule out arbitrage profits in the binomial tree (see the appendix to this chapter for a proof). It states that neither the stock’s nor the mma’s returns dominate the other, a necessary condition for an economic equilibrium—and this makes intuitive sense. If one security is superior to the other, why invest in the inferior asset? This common sense arbitrage-free condition is trivial to impose and easy to understand. It must also hold in the multiperiod binomial tree, and it can be generalized. For example, even if U, (1 + R), and D depend on the node in the tree, the same condition must hold for all nodes. This will prove useful in Part IV of the book, when we price interest rate options.

Stock Prices and Martingales Before pricing options, let us study the no-arbitrage condition to see if we can obtain any additional insights. Suppose that U > (1 + R) > D holds in the stock price tree so that arbitrage is ruled out. Algebra tells us that U > (1 + R) > D holds if and only if we can find a unique number 𝜋 strictly between 0 and 1 such that (1 + R) = 𝜋U + (1 − 𝜋) D

(17.4)

(You can see this intuitively: as U is large and D is small, you can alter 𝜋 to create any number that lies between them.) Dividing both sides of Equation 17.4 by (1 + R) and multiplying by S gives S=

𝜋US + (1 − 𝜋) DS 1+R

(17.5)

This expression has a useful interpretation. Look at the stock tree in Figure 17.1. Suppose we say that the “probability” of the stock going up is 𝜋, and that of going down is (1 – 𝜋). Then, the stock’s expected payoff computed with these probabilities and discounted to the present by the riskless rate (1 + R) is today’s stock price! This

THE SINGLE-PERIOD MODEL

gives us a simple method for computing the stock’s present value that is consistent with no arbitrage. Given a stock price tree, it is easy to determine 𝜋. Indeed, solving Equation 17.4 gives (1 + R) − D U − (1 + R) 𝜋= and (1 − 𝜋) = (17.6) U−D U−D What have we done? Our no-arbitrage condition implies the existence of some numbers, 𝜋 and (1 – 𝜋), which we have interpreted as probabilities, and we have used them to compute the stock’s present value. However, it is important to emphasize that they are not the actual probabilities of the stock going up or down (which are q = 3/4 and [1 – q] = 1/4 in this case), so we call them pseudo-probabilities (or martingale probabilities or risk-neutral probabilities). The last two names will make sense shortly. Next we rewrite Equation 17.5 two more times. Why? Because after it is rewritten, it is easier to see how the binomial model relates to the BSM OPM studied in Chapter 19. We thus rewrite Equation 17.5 as (with T = 1)

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S = E𝜋 [S (T)] e−rT

(17.7a)

where E𝜋 (.) is shorthand for denoting that we are computing an expected value by using the probabilities 𝜋 and (1 – 𝜋), and we replaced (1 + R) with erT . This method of computing the stock’s present value as its expected payoff using the pseudo-probabilities and discounting it backward through time with the mma’s value (expression [17.7a]) is also known as martingale pricing. To see this interpretation, we need to rewrite this equation using the notation for a mma’s value. Let A and A(1) = er be the time 0 and time 1 value, respectively, of the mma. Then, Equation 17.7a is S (T) S = E𝜋 (17.7b) [ A (T) ] A This expression shows that S(t)/A(t) is a martingale under the pseudo-probabilities. What is a martingale? It’s a stochastic process X(t) whose time t value equals the expected value of X(T) at some later time T. Martingales are associated with “fair gambles” because what you have today is what you expect to have tomorrow. In finance, this sense of fairness gives us a pricing system with no arbitrage opportunities. Because of Equation 17.7b the pseudo-probabilities are often called the martingale probabilities.

The Pricing Model Inspired by the example, we create a synthetic call using a replicating portfolio V (0) [≡ mS + b] with m shares of the stock worth S per share and b units of the mma priced at a dollar each. At time 1, we want to find (m, b) to match the portfolio’s value to the traded call’s value in each state. This yields two equations in two unknowns: in the “up” state ∶

V(1)UP = mUS + b (1 + R) = cU

(17.8a)

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in the “down” state ∶

V(1)DOWN = mDS + b (1 + R) = cD

(17.8b)

where cU = max (US − K, 0) and cD = max (DS − K, 0)

(17.8c)

A solution exists and solving them gives m = (cU − cD ) / (US − DS) b = (UcD − DcU ) / [(U − D) (1 + R)]

(17.9a) (17.9b)

m is known as an option’s delta or the hedge ratio. We will soon show why it is the holy grail of options pricing. The cost of constructing this portfolio is V (0) ≡ mS + b cU − cD UcD − DcU S+ (U − D) S (U − D) (1 + R) (1 + R) − D U − (1 + R) 1 c + c = 1 + R [( U − D ) U ( U − D ) D ]

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=

(17.10)

Now to prevent arbitrage, the synthetic call’s cost of construction, V (0), must equal the traded call’s price, c. This is the option’s arbitrage-free price! As seen in Equation 17.10, this value depends only on (S, U, D, R, K), which we refer to somewhat flippantly as KRUDS. These are the current stock price (S), the up and down factors (U, D), the dollar return on the mma (1 + R), and the contract’s strike price (K). Missing from the call’s value are q and (1 – q), the actual probabilities of the stock going up or down. This omission is important and it occurs because, in the derivation of the option’s price, we use an exact replication argument. No matter which node of the tree the stock price moves, the synthetic call’s payoffs exactly duplicate the option’s payoffs. Hence we don’t care what the actual probability is of moving up or down. The omission of the actual probabilities from the option’s price is important because it implies that if two investors agree on the KRUDS, but they disagree on q, they will still agree on the call’s price. Of course, disagreement about the probability q might lead to disagreement about the stock price S when (U, D, R) are fixed. But if agreement about S occurs (which you can readily see for actively traded liquid stocks), then the call’s price is fixed by our no-arbitrage argument.

The Hedge Ratio Classical myths depict the Holy Grail as the cup used by Jesus Christ at the Last Supper. Believed to possess miraculous powers, it has been lost and sought for centuries. Indeed, the quest for the lost Holy Grail features prominently in the tales of the legendary knights of King Arthur’s court. Options pricing theory also has its holy grail, the hedge ratio, the number of shares of the stock to hold for each written call to form a perfect hedge. This hedge ratio has miraculous powers because with it one

THE SINGLE-PERIOD MODEL

can construct a synthetic call to obtain the arbitrage-free price of a traded option. Without it, our argument fails. The hedge ratio for the binomial model is given by Equation 17.9a: m=

cU − cD (U − D) S

To prove this, consider a portfolio consisting of the short call and m shares of the stock. The initial value of this portfolio is − c + mS The values of this portfolio at time 1 in the up and down states are: cU D − cD U (U − D) cU D − cD U − cD + mDS = (U − D) − cU + mUS =

Because these values are equal, the covered call is riskless—and we have found the holy grail! To reemphasize, the hedge ratio works perfectly if the world behaves according to our single-period model. Finding a hedge ratio that works for more complex and realistic OPMs is a much harder task. Our quest for the holy grail (of options pricing) continues in the coming chapters for these more complex OPMs.

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Risk-Neutral Valuation Another interesting insight can be obtained using some additional algebra on Equation 17.10. We can use the pseudo-probabilities to rewrite the call’s value in another form, which has a nice economic interpretation. Recall that Equation 17.6 gave 𝜋 = [(1 + R) – D]/(U – D) and (1 – 𝜋) = [U – (1 + R)]/(U – D). Using them, we can rewrite the call’s arbitrage-free value (17.10) as c = [πcU + (1 − π) cD ] / (1 + R)

(17.11)

This equation is an example of the risk-neutral valuation procedure. The option’s value is its expected payoff using the pseudo-probabilities and discounted using the mma’s rate. As the legendary baseball coach Yogi Berra said, “it’s like déjà vu, all over again!” The pseudo-probabilities 𝜋 and (1 – 𝜋) are back to give us a simple way of computing present values! This interpretation gives the pseudo-probabilities an alternative name: risk-neutral probabilities. Using Equation 17.8c, we can write (17.11) compactly (using T = 1) as c = E ᄡ {max [0, S (T) − K]} e−rT

(17.12)

which will be shown later to be similar to an expression used to express the BSM model in Chapter 19.

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EXAMPLE 17.2: Risk-Neutral Pricing of European Options in a Single-Period Binomial Model ■

Consider the data from Example 17.2. The model starts at date 0 (time 0) and ends at date 1 (time T).

■

Three securities trade: - YBM’s stock price S is $100 today. The up factor U is 1.2000 and the down factor D is 0.9025.3 - A dollar invested in the mma earns r = 5 percent per year continuously compounded and gives a dollar return of (1 + R) = erT = e0.05 × 1 = $1.0513 in one year. - As the strike price K is $110, the option prices at maturity are cU = max(120 – 110, 0) = 10.00 and pU = max(110 – 120, 0) = 0 when the stock goes up to $120.00 and cD = max(90.25 – 110, 0) = 0 and pD = max(110 – 90.25, 0) = 19.75 when the stock goes down to $90.25.

■

Equation 17.6 gives the pseudo-probability of an up movement 𝜋 = [(1 + R) – D]/(U – D) = 0.5001.

■

The risk-neutral pricing equation (17.9) gives today’s option prices: c = [𝜋cU + (1 − 𝜋) cD ] / (1 + R) = $4.76 p = [𝜋pU + (1 − 𝜋) pD ] / (1 + R) = $9.39

These are the same prices that we got in Example 17.2.

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3

We derive them by using 𝜎 = 0.142470 in Jarrow–Rudd specification (see Extension 18.1).

Actual versus Pseudo-probabilities We noted that the pseudo-probabilities (𝜋 and [1 – 𝜋]) for up and down movement in the stock prices, respectively, differ from the actual probabilities (q and [1 – q]). But how are they related? Because the two sets of probabilities denote positive fractions, we can link them by introducing two positive terms 𝜙U and 𝜙D that denote an adjustment for risk: 𝜋 = 𝜙U q

(17.13a)

(1 − 𝜋) = 𝜙D (1 − q)

(17.13b)

The appendix to this chapter shows why 𝜙U and 𝜙D are an adjustment for risk. We note that 𝜙U and 𝜙D are not independent. Knowing one, the other can be computed since 𝜙U q + 𝜙D (1 – q) = 1. The intuition is simple. To compute present values, it is well known that one can take expected future values (using the actual probabilities) and discount to the present using a risk-adjusted rate. The larger the risk, the larger is the risk-adjusted discount rate. However, if one uses the pseudo-probabilities instead to compute a present value as in the risk-neutral valuation Equation 17.11, then the discount rate becomes the riskless rate. Because risk-neutral valuation does not adjust the discount rate for risk,

SUMMARY

to get the same present value, the adjustment for risk must occur in the use of the pseudo-probabilities (as distinct from the actual probabilities). This is indeed the case. Building on this insight, the appendix also discusses how these two sets of probabilities and the risk premium can be estimated using market data. We note that each set of probabilities has its own use: the actual probabilities contribute (in conjunction with historical data) to the estimation of the volatility, while the pseudo-probabilities (which we manufacture by using the risk-free interest rate, the length of a time interval, and the volatility; see Extension 18.1) are used for computing option prices. These actual and pseudo-probabilities will stay with us in the chapters that follow.

17.7

Summary

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1. In 1973, Fischer Black, Myron Scholes, and Robert Merton revolutionized the field of derivatives pricing by publishing the BSM model. Since 1973, options pricing developed along four, somewhat interlinked directions: (1) improving the BSM model and using it for different underlyings, (2) developing models that allow for random interest rates, (3) putting derivatives pricing on a rigorous mathematical foundation, and (4) using computational methods, including the binomial framework. 2. The binomial model has several advantages: (1) it is a great teaching aid that gives a simple, intuitive introduction to options pricing, (2) it can be repeated to build a multiperiod tree, which gives prices that closely approximate the BSM model, (3) it utilizes martingale pricing, the paramount technique of derivatives valuation, and (4) it is a versatile model that prices standard European and American options and other derivatives. 3. The binomial OPM makes some key assumptions: no market frictions, no credit risk, competitive and well-functioning markets, no interest rate uncertainty, no intermediate cash flows, and no arbitrage. In addition, it assumes that the underlying trades at discrete time intervals and follows a binomial process. 4. The OPM constructs a synthetic option that replicates a traded option’s payoff. Next, if no arbitrage opportunities exist, the two must have the same value. An option’s arbitrage-free price is the cost of constructing the synthetic option. 5. The binomial OPM is set up as follows: - Three securities trade: a stock, a riskless mma, and a European (call or put) option. - The stock price goes up and down by the factors of U or D, respectively. - A dollar invested in the mma gives a dollar return of (1 + R) after one period. - As the strike price is given, the option’s payoffs at maturity are determined by the stock prices on the tree. Solve for the option price by the formula Option price = [𝜋 × (Option payoff)UP + (1 − 𝜋) × (Option payoff)DOWN ] / (1 + R)

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where 𝜋 = [(1 + R) – D]/(U – D) is the pseudo-probability of an up movement, and option payoffs “up” and “down” are considered at the up and down nodes. - The method of computing the expected payoff with pseudo-probabilities and discounting with the mma’s value is known as risk-neutral valuation.

17.8

Appendix

Proving the No-Arbitrage Argument The no-arbitrage condition U > (1 + R) > D proves useful in many contexts. 1. Single-period binomial model: U > (1 + R) > D (“Bounded Dollar Return”): This is a necessary and sufficient condition for no arbitrage in a single-period tree. This is the result proven in this extension. 2. Multiperiod binomial model: This result must hold at every node in the multiperiod binomial tree. 3. Binomial interest rate derivatives pricing model: A generalized version of this result is required for pricing interest rate derivatives in a binomial framework: if U, (1 + R), and D depend on the node in the tree, then the bounded dollar return condition must hold for all nodes (see Jarrow, 2002b).

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4. Pseudo-probabilities: We will soon see that this condition is intimately connected with the existence of pseudo-probabilities (or equivalent martingale probabilities or risk-neutral probabilities) in binomial models. Thus the condition that we establish is linked to the core ideas at the heart of options pricing.

RESULT 1 No-Arbitrage Condition The condition U > (1 + R) > D is equivalent to no arbitrage.

PROOF OF PART I: NO ARBITRAGE IMPLIES U > (1 + R) > D. Assume the condition U > (1 + R) > D fails. We show that this gives arbitrage profits. A violation of U > (1 + R) > D can happen in one of two ways—either (1 + R) is to the left side of U > D (i.e., [1 + R] ≥ U > D) or to its right (i.e., U > D ≥ [1 + R]). ■

Suppose (1 + R) ≥ U > D. This makes the mma a better investment than the stock, irrespective of the stock’s going up or down. Suppose the stock price is $100 today. Short sell the stock and invest the 100 proceeds in the mma. At the end of the period, your arbitrage profit is 100[(1 + R) – U] ≥ 0 if the stock goes up or 100[(1 + R) – D] > 0 if the stock goes down. This is an arbitrage strategy.

APPENDIX

■

Suppose U > D ≥ (1 + R). This makes the mma an inferior investment to the stock. Borrow at the risk-free rate by short selling the mma, invest the proceeds in the stock, and collect arbitrage profits.

PROOF OF PART II: U > (1 + R ) > D IMPLIES NO ARBITRAGE. Recall from chapter 6 that arbitrage can be expressed as a costless strategy that nonetheless pays something in the future. With two assets, a stock, and a mma, how would you obtain a zero-investment portfolio? Obviously, you cannot buy or sell both assets simultaneously (how would you finance such trades?). You must buy one asset and short the other. However, you cannot make arbitrage profits if U > (1 + R) > D holds. Why? Because in one of the up or down nodes, the payoff to the portfolio will be negative. To see this, suppose that you buy the stock and short the mma obtaining a zero initial investment portfolio. The payoff at time 1 if you go up is U – (1 + R) > 0. The payoff if you go down at time 1 is D – (1 + R) < 0. This portfolio is not an arbitrage opportunity. Conversely, if you short the stock and long the mma, the zero initial investment portfolio has a negative payoff in the up node and a positive payoff in the down node. This zero investment portfolio is also not an arbitrage opportunity.

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The Probabilities and Risk Premium RELATING ACTUAL AND PSEUDO-PROBABILITIES Let us further explore the concepts of the actual probabilities (q, 1 – q) and the pseudo-probabilities (𝜋, 1 – 𝜋) in the context of our single-period model and compute the risk premium that connects them. First, the probabilities must agree on zero-probability events. If one of the actual probabilities is zero, the stock price tree will fail to branch out and the stock becomes a riskless asset. The condition q > 0 if and only if 𝜋 > 0 prevents this collapse. Next, recall that we linked the pseudo-and actual probabilities by Equations 17.13a and 17.13b: 𝜋 = 𝜙U q (1a) (1 − 𝜋) = 𝜙D (1 − q)

(1b)

where 𝜙U and 𝜙D are positive numbers that adjust for risk. We note that 𝜙U and 𝜙D are not independent. Knowing one, the other can be computed since 𝜙U q + 𝜙D (1 – q) = 1. You can write them as two values taken by the random variable 𝜙.̃ To summarize, today’s stock price is S, and the parameters take on the following values after one period: S (1) =

US is associated with q > 0, 𝜋 > 0, and 𝜙̃ = 𝜙U {DS corresponds to (1 − q) > 0, (1 − 𝜋) > 0, and 𝜙̃ = 𝜙D

It follows from this definition that the expectation of the random variable 𝜙̃ computed using the actual probabilities is given by E (𝜙̃ ) ≡ 𝜙U q + 𝜙D (1 − q) = 𝜋 + (1 − 𝜋) = 1

(2)

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THE RISK PREMIUM Equation 17.5 allows us to write the present value of the stock price at time 0 as the discounted expected payoffs using the pseudoprobabilities: 𝜋US + (1 − 𝜋) DS (3) S= 1+R Using (1a) and (1b), we rewrite this as S=

q (𝜙U US) + (1 − q) (𝜙D DS) 1+R

(4)

Notice that we are adjusting for risk via the cash flows in the numerator. This contrasts with the traditional way of computing a present value. The traditional method is to adjust the denominator. You may have taken courses in capital budgeting or project evaluation, in which you learned to compute the net present value by discounting risky cash flows, positive or negative, with a risk-adjusted discount rate. Interestingly, 𝜋 and q are related by a risk premium. Using algebra and statistics, we show that the risk premium is given by cov(−𝜙,̃ S(1)/S ). To establish this, first, note from Equation 17.17 that we have S=

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or 1 + R =

̃ (1)) E (𝜙S 1+R ̃ (1)) E (𝜙S S

(5)

where E(.) denotes the expectation computed by using the actual probabilities q and (1 – q). A familiar result in statistics (for two random variables x and y, cov[x, y] = E[xy] – E[x]E[y]) gives ̃ (1)) − E (𝜙̃ ) E (S (1)) cov (𝜙,̃ S (1)) = E (𝜙S

(6)

Using (2) and transferring the last expression to the left-side, we get ̃ (1)) cov (𝜙,̃ S (1)) + E (S (1)) = E (𝜙S

(7)

We can write (5) using (7) and another result from statistics [cov(ax, y) = acov(x, y)] as

1+R=

̃ E (𝜙S(1)) S (1) E (S (1)) = cov 𝜙,̃ + ( ) S S S

(8)

APPENDIX

Rearrangement of Equation 8 gives E (S (1)) S (1) S (1) = 1 + R − cov 𝜙,̃ = 1 + R + cov −𝜙,̃ ( ( S S ) S ) E (S (1)) or S = S (1) 1 + R + cov −𝜙,̃ [ ( S )]

(9)

We rewrite Equation (9) as the following result.

RESULT 2 The Risk Premium Linking Actual and Pseudo-probabilities In a single-period model, the actual and pseudo-probabilities are linked by a risk premium cov(−𝜙,̃ S(1) / S), which is given in the following expression:

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S=

q (US) + (1 − q) DS S (1) 1 + R + cov − 𝜙̃ , ( S )

(10)

where cov( , ) is the covariance operator.

Thus the risk premium is cov(−𝜙,̃ S(1)/S). As shown, the risk premium depends explicitly on the pseudo-probabilities, proving they give an adjustment for risk. The risk premium can be explicitly derived in an equilibrium model. Equilibrium models (such as the capital asset pricing model) require the equality of demand and supply, the presence of utility functions and endowments, which determine risk premium (defined as the expected risky return less the risk-free rate). By contrast, arbitrage models (like our options pricing models) do not need such features and use only the no-arbitrage condition to derive the results. Notice that the difference in the two approaches to computing present values is that we only need to use arbitrage pricing for expression (3), but for expression (10), we need to compute both 𝜙U and 𝜙D , which requires an equilibrium model.

ESTIMATING PROBABILITIES How would you compute the probabilities? As noted before, the computation of the pseudo-probabilities (𝜋, 1 – 𝜋) is given in Equation 17.6. The actual probabilities can be computed using historical data and

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statistics for a single-period binomial distribution. First, using historical data, estimate the expected return Ê (return); then set it equal to the theory’s expected return, Ê(return) = qU + (1 – q)D, and solve for q. Remember that this is a toy model, and although we have shown how to solve for q, the estimation issues become more involved in the case of more complex models—but they use the same ideas as shown here!

17.9

Cases

Leland O’Brien Rubinstein Associates Inc.: Portfolio Insurance (Harvard Busi-

ness School Case 294061-PDF-ENG). The case studies the rise and fall of Leland O’Brien Rubinstein Associates’ portfolio insurance selling business. Leland O’Brien Rubinstein Associates Inc.: SuperTrust (Harvard Business

School Case 294050-PDF-ENG). The case examines Leland O’Brien Rubinstein Associates’ attempts to rebuild itself after the 1987 stock market crash by creating new products to meet the unsatisfied needs of equity investors. Tata Steel Limited: Convertible Alternative Reference Securities (B) (Richard

Ivey School of Business Foundation Case 910N32-PDF-ENG, Harvard Business Publishing). The case considers valuation of a convertible bond offering by a global top ten steel producer.

17.10 Questions and Problems Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

17.1. Why is the binomial model a useful technique for approximating options

prices from the Black–Scholes–Merton model? Describe some applications and uses of this model. 17.2. How is the no-arbitrage principle used in the binomial model to find options

prices? 17.3. In the binomial options pricing model, it is assumed that the stock price

follows a binomial process. a. Is this a reasonable description of the actual stock price process? b. If not, why should one study this model? 17.4. a. In the binomial options pricing model, what assumptions are made about

dividends and interest rates? b. In the binomial stock price tree, what restrictions are needed on the up

and down factors, relative to the risk-free rate, to avoid arbitrage? Explain. 17.5. In the binomial options pricing model, what is the hedge ratio? 17.6. a. In the binomial options pricing model, what are the pseudo-probabilities? b. In the binomial options pricing model, what does risk-neutral valuation

mean? Explain.

QUESTIONS AND PROBLEMS

17.7. When pricing an option using risk-neutral valuation, one is assuming that all

investors are risk neutral. Hence, if one believes that investors are risk averse, risk-neutral valuation cannot be used. True or false? Explain your answer.

FIGURE 17.4: Today

After 1 year 120.773898

100 89.47150422 Call: strike price K = 105, option matures after 1 year. Money Market Account (mma)

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1

1.051271096

Note: The up factor U is US/S = 120.773898/100 = 1.2077. The down factor D is DS/S = 89.471504/100 = 0.8947 Dollar return 1 + R = 1.0513. We report numbers to four places and the final answer to two places after the decimal point although calculations consider ten places after the decimal point. The next eight questions use the above data. 17.8. a. Given the preceding data, set up a perfect hedge and compute the call

option’s value. b. What is the hedge ratio? What does it signify? 17.9. a. What are the pseudo-probabilities? b. Show that the stock price is its discounted expected value using these

pseudo-probabilities. 17.10. Compute the call option’s value using risk-neutral valuation. 17.11. Suppose the market price of the call option is $6. Is there an arbitrage

opportunity? Show how you can take advantage of this price to make arbitrage profits. 17.12. a. Set up a perfect hedge and compute the put option’s value. Let the put

have the same strike price K = $105. b. What is the hedge ratio? What does it signify?

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17.13. Compute the put option’s value using risk-neutral valuation. 17.14. Using the data from questions 17.8 and 17.11, verify that put–call parity holds. 17.15. For the data in question 17.13, suppose the market price of the put option is

$3. Is there an arbitrage opportunity? Explain how one would trade to exploit this arbitrage. The next four questions are based on the following data for a single-period binomial model: ■

A stock’s price S is $50. After six months, it either goes up by the factor U = 1.22095341 or it goes down by the factor D = 0.79881010.

■

Options mature after T = 0.5 year and have strike price K = $45.

■

A dollar invested in the money market account earns continuously compounded risk-free interest at 2 percent per year.

17.16. Compute the call’s value. 17.17. Suppose the market price of the call option is $5. Is there an arbitrage

opportunity? Explain how one would trade to exploit this arbitrage. 17.18. Consider the following exotic option whose payoff at expiration is given

by the stock price squared less a strike price if it has a positive value, zero otherwise:

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max [S(1)2 − K, 0] Assuming that the strike price K is $2,500, determine the value of this exotic option under the assumption of no-arbitrage. If the market price of the call is $600, how would you trade to exploit this arbitrage opportunity? 17.19. Consider the following exotic option whose payoff at maturity is given by

the square root of the stock price less the strike price if it has a positive value, zero otherwise: max [√S (1) − K, 0] Assuming that the strike price K is $7, determine the value of this exotic option under the assumption of no-arbitrage. If the market price of the call is $0.10, how would you trade to exploit this arbitrage opportunity? 17.20. (Microsoft Excel) Implied Volatility

Consider the following data for computing option prices given to you by your professor.

QUESTIONS AND PROBLEMS

FIGURE 17.5: Stock Today

After 1 year

120.77389800 100 89.47150422

Call: strike price K = 105, option matures after 1 year. Money Market Account (mma)

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1

1.051271096

You want to know how this data was generated. The professor, when asked, apologizes and says, “I used a Jarrow-Rudd approximation but I lost the data. You are an Excel expert—why don’t you use ‘Goal Seek’ and determine what the volatility is?”

391

18 Multiperiod Binomial Model Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

18.1 Introduction

Recasting the Two-Period Example in the Multiperiod Framework

18.2 Toward a Multiperiod Binomial Option Pricing Model

The n-Period Binomial Option Pricing Model

The Stock Price Evolution

EXTENSION 18.1 Linking the Binomial Model to the Black–Scholes–Merton Model

Binomial Option Price Data The Stock Price Tree

18.3 A Two-Period Binomial Model 18.5 Extending the Binomial Model Backward Induction

Known Dividends

Option Pricing via Synthetic Construction (Method 1)

Valuing American Options

Repeat, Repeat: Risk-Neutral Pricing (Method 2)

18.6 Spreadsheet Applications A Two-Period Binomial Example

One-Step Valuation: Prelude to the Multiperiod Model (Method 3) 18.7

18.4 The Multiperiod Binomial Option Pricing Model Binomial Coefficients and Pseudo-probabilities

A Sixteen-Period Example

Summary

18.8 Cases 18.9 Questions and Problems

TOWARD A MULTIPERIOD BINOMIAL OPTION PRICING MODEL

18.1

Introduction

Not too many years back, a large US consulting company valued some options for a client. Two groups within the company were given the job. One of them used the celebrated Black–Scholes–Merton (BSM) option pricing model (OPM), which gives a neat analytical solution. This team (involving one of us) argued that the BSM model is useful because a formula enables the computation of changes in the option price in response to changes in the stock price (later, you will see how this helps traders to hedge and manage an options portfolio). The rival group used a multiperiod binomial OPM with thirty time periods and thirty-one branches in an Excel spreadsheet. They argued that the binomial model is easier for clients to understand and that it gives prices close to the BSM values. Both claims are correct, yet ease of understanding carried the day. This chapter extends the binomial model to a multiperiod setting and shows how to set the parameters to get prices that closely approximate the BSM. First, we develop a two-period binomial OPM repeating concepts and methods used earlier. Riskneutral valuation, when generalized, takes us to the multiperiod model. We also show the versatility of this numerical approximation technique when it is used to price American options and for including dividends. Finally, we discuss setting up the binomial model in a spreadsheet program like Microsoft Excel.

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18.2

Toward a Multiperiod Binomial Option Pricing Model

To construct the multiperiod binomial OPM, we first need to understand the assumed stock price process.

The Stock Price Evolution Figure 18.1 shows how a stock price path looks under the binomial assumption. The first figure shows the path in the single-period binomial model, which is the model from chapter 17. It’s a line parallel to the horizontal axis. Here it corresponds to the value US. Alternatively, it could have taken the value DS. In looking at this graph, one can easily reject the single-period model as a good approximation over, say, a one-year horizon when compared to the actual stock price path given in the next figure. The next figure also shows the stock price path in a multiperiod binomial setting. You can still reject this evolution just by observation, but what happens if these periods shrink and become really small? We have superimposed a realistic looking stock price path on this graph. When the time steps become small, the multiperiod binomial price process approximates this actual price path quite well! Here the multiperiod binomial model may not be so easily rejected. In fact, choosing the actual probability of going up q, the up factor U, and the down factor D appropriately, the multiperiod binomial evolution approaches the lognormal distribution that we use in the next chapter to obtain the BSM model.

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FIGURE 18.1: The Stock Price Evolution in a Single Period and Multiperiod Setting Single Period Stock price Realized stock price path

US S DS

Possible path

0 Today

After 1 year

Time

Multiple Periods Stock price Actual stock path Stock price path (model)

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U2S US S DS D2S

0 Now

1 year

Time

Binomial Option Price Data Our examples will use the binomial option pricing data (BOP data!) for Your Beloved Machines Corp. (YBM), which are the same data used in Example 17.1 in the last chapter. ■

YBM’s current stock price S is $100.

■

We consider a European call on YBM with strike price K = $110 and maturity time T = 1 year.

■

The model starts today (time 0) and ends after one year at time T = 1.

■

The stock price return volatility 𝜎 is 0.142470.

■

The number of periods n varies; it can be 1, 2, 3, or more. If n is 2, we divide the time to maturity T into two periods of length Δt = T/n = 0.5 year each. If n is 3, then Δt = 1/3 = 0.3333 year, and so on.

TOWARD A MULTIPERIOD BINOMIAL OPTION PRICING MODEL

■

We let the continuously compounded risk-free interest rate r be 5 percent per year. In the one-period model, the dollar return 1 + R = e0.05 = $1.0513 after one year. In the two-period model, the dollar return 1 + R = erΔt = e0.05×0.5 = $1.0253 after one period and (1 + R)2 = 1.02532 = $1.0513 after two periods—at year’s end. This is a benefit of continuous compounding. No matter how you slice and dice the time intervals, your investment still grows by the same amount over a year. Recall that all of our computations are performed to 16 decimal place accuracy, although for clarity in exposition, we report rounded numbers up to either 2 or 4 decimal places.

The Stock Price Tree The next step is to price options in a two-period binomial model. Example 18.1 presents the stock price tree and the money market account used for options pricing.

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EXAMPLE 18.1: A Stock Price Tree and Option Prices at Maturity ■

As before, a stock, a money market account, and a European call option trade. We use the BOP data. The model has three dates: today is the starting date (time 0), six months later is date 1, and the maturity is date 2 (time T = 1 year), when the call expires. As the number of periods n is 2, there are two time periods of length Δt = T/n = 0.5 years each.

■

The asset price evolution is shown in Figure 18.2. The dollar return is 1 + R = $1.0253 after one period. YBM’s stock price evolves according to a multiplicative binomial model. From any node in the binomial tree, the next period’s stock price is obtained by either multiplying the current stock price by the up factor U or by the down factor D. We compute these factors using specification 1 in Table 18.1 (see Extension 18.1): U = 1.1282 and D = 0.9224. - Starting from a price of S = $100 at date 0, the stock can either go up to US = 1.1282 × 100 = $112.82 or down to DS = $92.24 at the end of the first period. - If the stock price is US at date 1, then it either goes up to U(US) or down to D(US) at date 2. If it has a value of DS at date 1, then it subsequently goes up to U(DS) or down to D(DS). Hence U (US) = 1.1282 × 112.82 = $127.29 D (US) = 0.9224 × 112.82 = $104.07 = U (DS) D (DS) = 0.9224 × 92.24 = $85.08 - Notice that the “up and then down” dotted path combines with the “down and then up” dashed route because D(US) = U(DS). The branches combine because we use a multiplicative model. This feature is very useful when there are many periods. For example, if the branches recombine, a twoperiod model would have three instead of four separate nodes. However, in a 10-period model, there would be 10 + 1 = 11 separate nodes in a recombining tree but 210 = 1,024 separate nodes if the branches do not recombine. - A generalization of the result (U > 1 + R > D) of chapter 17 is essential to make the tree arbitragefree. We assume that this condition holds at every node in the tree.

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- The actual probability of the stock going up is q = 3/4 and that of YBM going down is (1 – q) = 1/4. Multiplying the probabilities together gives the probability of two up movements as q2 = 9/16, an up and a down movement as 2q(1 – q) = 3/8, and two down movements as (1 – q)2 = 1/16. ■

You can easily compute the options prices at date 2. - The call’s value at maturity is given by max[0, S(2) – K]. Consider the topmost node in Figure 18.2. The stock price there is U 2 S. Thus when the stock goes “up and then up,” the call value at time 2 is c(2)UP,UP ≡ cUU = max(U2 S – K, 0). Note that “UU” appears as a subscript in the call price. Also note that we drop “(2)” to reduce clutter. The call has a strike price K = $110. Hence cUU = max(127.29 – 110, 0) = 17.29. The call finishes out-of-the-money at the other two nodes on date 2, so c(2)UP,DOWN ≡ cUD = 0 and c(2)DOWN,DOWN ≡ cDD = 0. - The put’s value at date 2 is max[K – S(2), 0]. Using this formula, pUU = $0, pUD = $5.93, and pDD = $24.92.

18.3

A Two-Period Binomial Model

We will utilize the previous data to compute European options prices using a technique called backward induction.

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Backward Induction In Sir Arthur Conan Doyle’s first novel A Study in Scarlet (1887), Sherlock Holmes made a classic comment about his crime detection technique: “In solving a problem of this sort, the grand thing is to be able to reason backward. That is a very useful accomplishment, and a very easy one, but people do not practise it much.” Interestingly the approach used by Sherlock Holmes is called backward induction, a mathematical technique where one starts at the end and then solves a problem by working backward through time. We use it to solve the multiperiod binomial option pricing. The steps are as follows: ■

Create the binomial stock price tree by repeatedly attaching the single-period structure to the end of each node until the tree reaches the option’s maturity date.

■

Starting at the last date in the tree, given the strike price, you can easily compute the option’s values on the expiration date for all possible final nodes.

■

At the end of the last period, consider the option values ordered from the top node to the bottom node. Select the top two option values and apply the risk-neutral pricing formula to get the topmost option price at the previous date. Next, move down one node at the final time, and repeat this computation for the next two option values. Continue moving down the final nodes of the tree in this fashion, until the bottommost node is reached. When reached, this process generates all the option prices at the beginning of the last period from the top to the bottom of the tree.

A TWO-PERIOD BINOMIAL MODEL

FIGURE 18.2: Two-Period Binomial Model for Pricing a European Option Numerical Example Stock Today

Stock 6 months

Stock 1 year

Call 1 year

Put 1 year

Pseudo Prob Actual Prob 1 year 1 year

127.29

17.29

0

0.25

9/16

104.07

0

5.93

0.50

6/16

85.08

0

24.92

0.25

1/16

Stock Date 2

Call Date 2

Put Date 2

Pseudo Prob Actual Prob Date 2 Date 2

U2S

cUU

pUU

π2

q2

UDS

cUD

pUD

2π (1 – π)

2q (1 – q)

D2S

cDD

pDD

(1 – π)2

(1 – q)2

112.82 100.00 92.24

Two-Period Model Stock Date 0

Stock Date 1 π π

S (1 – π) (1 – q)

q

US (1 – π) q DS (1 – q)

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Money Market Account (mma)

■

Date 0

Date 1

Date 2

1

1.0253 1

1.0513 1.0253

Today

6 months

1 year

1

1+R

(1 + R)2

Step back one time step (toward date 0), and compute the option values at the beginning of the current period, using the procedure just described. Continue doing this period by period, working backward through time. Eventually you come to today and then the process ends. The solution is today’s option price.

Option Pricing via Synthetic Construction (Method 1) This section uses data from Example 18.1 to demonstrate how to price options in a two-period binomial framework. We begin by setting up a dynamic hedge using the stock and a money market account to replicate the call’s value. Dynamic hedging is the extension of our synthetic construction to a multiperiod setting.

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

We use a dynamic, self-financing trading strategy to construct a portfolio that replicates the traded option’s payoffs. This portfolio is dynamic because it changes over the multiple periods, and it is self-financing because there are no net cash flows at intermediate dates. When no arbitrage is allowed, the cost of constructing this synthetic option must equal the price of the traded option.

EXAMPLE 18.2: Pricing a Call via Synthetic Construction ■

When solving the model, like Sherlock Holmes, we have to work backward from the end. Inspired by the example, we create at date 1 (one date before the last date) a portfolio V (1) ≡ m(1)S(1) + b(1) with m(1) shares of the stock worth S(1) and b(1) units of the money market account priced $1 each. This portfolio is constructed to replicate the option’s values at date 2. It is the synthetic call option. Its cost of construction must equal the traded call value at date 1, or else arbitrage opportunities arise.

■

We have to do the replication exercise twice: once when the stock goes up to US and once more when it goes down to DS. In each case, we have to solve two equations in two unknowns to find the share holdings m(1), b(1).

■

We use the same data as in Example 18.1 (see Figure 18.2 for a summary of this information) to illustrate this method. To reduce clutter, we use U and D in the subscripts instead of UP and DOWN, for example, V (2)UP,UP is expressed as V (2)UU ⋅ At date 2, when the stock is at US (= 112.82) on date 1, In the “up, up” state, V (2)UU = m (1) U2 S + b (1) (1 + R) = cUU

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In the “up, down” state,

V (2)UD = m (1) UDS + b (1) (1 + R) = cUD

Solving these equations, we get m(1)U = (cUU – cUD ) / (U2 S – UDS) = 17.29/ (127.29 – 104.07) = 0.7445 b(1)U = (UcUD – DcUU ) / [(U – D) (1 + R)] = – (0.9224 × 17.29) / [(1.1282 – 0.9224) 1.0253] = –75.57

(18.1)

Thus the cost of constructing the synthetic call at date 1 in the up state is V (1)U = m(1)U US + b(1) = 8.4345 = $8.43. To rule out arbitrage, this must equal the traded call’s price cU on date 1 in the up state. At date 2, when the stock is at DS (= 92.2363) on date 1, either solve the two equations to find that the call is worthless at date 1, or infer that if the call is sure to be worthless at date 2, then it must also be worthless at date 1. So when the stock goes down to 92.24, m(1)D = 0 and b(1)D = 0, giving the synthetic option a value of cD = 0 at date 1 in the down state. ■

Finally, let us now move back to date 0. Construct a portfolio V (0) ≡ m(0)S(0) + b(0) with m(0) shares of the stock worth S(0) and b(0) units of a money market account worth $1 each. This portfolio’s share holdings are constructed to match the synthetic call prices cU and cD at date 1, giving two equations in two unknowns. At date 1,

A TWO-PERIOD BINOMIAL MODEL

In the “up” state,

V (1)U = m (0) 112.82 + b (0) 1.0253 = 8.43 = cU

In the “down” state,

V (1)D = m (0) 92.24 + b (0) 1.0253 = 0 = cD

Solving these equations, we get m(0) = 0.4097 and b(0) = – 36.8536, so the cost of constructing the synthetic call at date 0 is V (0) = m (0) S (0) + b (0) = 4.1135 = $4.11 This must equal c(0) ≡ c, the traded call’s price today. Any other quoted price will create getrich opportunities for vigilant arbitrageurs. The numbers from this exercise are summarized in Figure 18.3.

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This simple example illustrates several important features of the multiperiod binomial OPM: ■

Prices differ from the single-period model prices. Despite using the same BOP data, the single- and two-period models give different options prices: $4.76 and $4.11, respectively. This is because the two trees aren’t identical. Compare the figures to notice that the stock and the option prices have different distributions at the maturity date.

■

Self-financing. We show next that the dynamic trading strategy is self-financing. Initiate the trading strategy at date 0 by buying 0.4097 units of the stock priced at $100 and finance this by selling 36.8536 units of the money market account worth $1 each. You still need to invest $4.11, which is the call price. Suppose the stock goes up to $112.82 after six months. Then the portfolio’s value is $8.43. Liquidate this portfolio and raise funds by selling 75.57 units of the money market account for $1 each. Use the total proceeds of $84 (= $8.43 + $75.57) to buy 0.7445 units of the stock. Obviously, this transaction is self-financing. After the up node, suppose the stock goes up again to $127.29 at the end of the year. Then the portfolio’s value is (0.7445 × 127.29 – 75.57 × 1.0253) = $17.29. If the stock goes down from there to $104.07, then the portfolio value is (0.7445 × 104.07 – 75.57 × 1.0513) = 0. These payoffs are identical to the call values at the corresponding nodes in the tree at date 2. Suppose the stock goes down to $92.24 at date 1 instead. Then the call becomes worthless. No trades are necessary. The call has zero value thereafter.

■

Dynamic market completion. Just two assets, a stock and a money market account, replicate the three option values at maturity. Note that we have dynamically completed the market, where the term complete refers to the portfolio’s ability to match all possible option values at maturity using a dynamic trading strategy.

■

A hedging interpretation. Notice the holy grail, the hedge ratios m(0), m(1)U , and m(1)D in the model. They tell us how to rebalance our portfolios to synthetically construct the option values in the nodes that follow.

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

FIGURE 18.3: The Two-Period Binomial Model for Pricing European Call Options (Numerical Values) Today (date 0)

c = 4.11 m(0) = 0.4097 b(0) = –36.8536 S = 100

6 months (date 1) cU = 8.43 m(1)U = 0.7445 b(1)U = –75.57 US = 112.82 cD = 0 m(1)D = 0 b(1)D = 0 DS = 92.24

1 year (date 2) cUU = 17.29 U2S = 127.29 cUD = 0 UDS = 104.07 cDD = 0 D2S = 85.08

$1 becomes 1 + R = $1.0253 after six months. Strike price K = $110.

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■

Independent of actual probabilities. Note that the actual probabilities of the stock moving up or down do not enter the valuation procedure. As in the single-period binomial model, this is because we exactly replicate the option’s payoffs at every node in the tree on the expiration date. Hence the probabilities of reaching these nodes do not affect the option’s values.

Repeat, Repeat: Risk-Neutral Pricing (Method 2) Why crank out option values by solving linear equations when we can summon the power of algebra? Plug the algebraic values from expression (18.1) into the portfolio’s value to get the synthetic call’s value at date 1: cU = V(1)U = m(1)U S(1)U + b(1)U =

cUU − cUD UcUD − DcUU US + (U’D) US (U’D) (1 + R)

Rearrange terms (keep (1 + R) in the denominator, gather cUU and cUD and rewrite) to get (1 + R) − D U − (1 + R) c + c ( U − D ) UU ( U − D ) UD cU = (18.2) 1+R Recalling Yogi Berra, “It’s like déjà -vu, all over again!” Our pseudo-probabilities are back. As before, we write the pseudo-probability of going up as 𝜋 = [(1 + R) – D]/(U – D) and going down as (1 – 𝜋) = [U – (1 + R)]/(U – D). Now you can compactly write the binomial call OPM as cU = [𝜋cUU + (1–𝜋) cUD ] / (1 + R)

(18.3)

A TWO-PERIOD BINOMIAL MODEL

401

One can develop similar formulas for call prices at the down node cD and at today’s date c. Moreover, replacing c with p gives the formula for pricing puts! Generalize this to get a formula for pricing European options.

RESULT 18.1 A Formula for Pricing Options by Repeated Application of Risk-Neutral Pricing c = [𝜋 × cU + (1–𝜋) × cD ] / (1 + R)

(18.4)

where 𝜋 = [(1 + R) – D]/(U – D), U and D are the up and down factors, respectively, and 1 + R is the dollar return. The same formula applies for puts.

The next example uses this formula.

EXAMPLE 18.3: Pricing a Call by Repeated Application of Risk-Neutral Pricing

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Using the same data as in Examples 18.1 and 18.2, we get 𝜋 = (1.0253 − 0.9224) / (1.1282 − 0.9224) = $0.50 cU = [𝜋cUU + (1 − 𝜋) cUD ] / (1 + R) = (17.29 × 0.50) /1.0253 = $8.43 (18.5)

cD = [𝜋cUD + (1 − 𝜋) cDD ] / (1 + R) = 0 c = [𝜋cU + (1 − 𝜋) cD ] / (1 + R) = (0.50 × 8.43) /1.0253 = $4.11 which verifies today’s call price.

One-Step Valuation: Prelude to the Multiperiod Model (Method 3) For pricing European options, you can simplify further and do it all in one step. In expression (18.4), plug the expressions for cU and cD into the call value c and simplify to get c=

𝜋cU + (1 − 𝜋) cD 𝜋2 cUU + 2𝜋 (1 − 𝜋) cUD + (1 − 𝜋)2 cDD = (1 + R) (1 + R)2

(18.6)

This is the two-period binomial model. The numerator computes the expected payoff using the pseudo-probabilities and the denominator discounts the cash flows. The model’s description follows:

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

■

As the pseudo-probability of going from S to US is 𝜋, and from US to U 2 S is also 𝜋, the pseudo-probability of going from S to U 2 S is 𝜋2 , which is the pseudo probability of reaching cUU . Similarly, the pseudo-probability of reaching UDS (which corresponds to the call price of cUD ) is 𝜋 (1 – 𝜋). As there are two such paths, multiply by 2, giving an overall probability of 2𝜋(1 – 𝜋). The stock payoff D2 S (and call value cDD ) has the pseudo-probability (1 – 𝜋)2 . These pseudo-probabilities are noted in the last column of the first figure.

■

Once you have these pseudo-probabilities, compute expected payoffs by multiplying each option payoff by the pseudo-probability of reaching those nodes.

■

Next, compute the overall expected payoff for the call by summing up the expected payoffs computed in the previous step.

■

This gives the expected payoff at time T. Discount by the dollar return (1 + R)2 to get today’s expected payoff.

EXAMPLE 18.4: A Two-Period Binomial Call Pricing Model Using the pseudo-probabilities from Example 18.3 and the option payoffs from Example 18.1 in expression (18.6) gives today’s call price:

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c = (0.25 × 17.29 + 0.50 × 0 + 0.25 × 0) /1.02532 = $4.11 One can also price European puts by any one of these three methods.1 Moreover, if you know the value of one option, you can price the other using put–call parity. 1

For example, to replicate a put, set up a portfolio V ≡ mS + b, as before. Working backward with the second-period put prices pUU = 0, pUD = $5.93, and pDD = $24.92, one gets pU = $2.89 and pD = $15.05 and, ultimately, today’s put price p = $8.75. Methods 2 and 3 work similarly.

EXAMPLE 18.5: European Put Pricing by Put–Call Parity Using the BOP data and the call price from the previous example, put–call parity for European options (Result 16.1) gives the put price as p = c + KB – S = 4.11 + 110 × (1/1.0513) –100 = $8.75

THE MULTIPERIOD BINOMIAL OPTION PRICING MODEL

18.4

The Multiperiod Binomial Option Pricing Model

We can utilize the Binomial Theorem from mathematics to develop a generalized version of the option pricing model. Many distinguished scholars have worked on this theorem including Sir Isaac Newton, truly the first rocket scientist!

Binomial Coefficients and Pseudo-probabilities The binomial coefficient (see the binomial theorem in Appendix A) provides a general formula for computing the number of ways of reaching a final payoff in the n multiperiod stock price tree. The binomial coefficient (read “n choose j”)gives (j) the number of combinations of selecting j objects out of n objects: n n! = ( j ) j! (n − j)!

(18.7a)

where the expression n! (read n-factorial) is defined as

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n! = n × (n – 1) × (n – 2) × … × 2 × 1 and 0! = 1

(18.7b)

Because the number of up movements uniquely identifies the number of different paths reaching a particular end-period stock payoff (which automatically fixes the number of down movements), we can use the binomial coefficient to compute the number of such paths. Suppose the stock price at expiration is Uj Dn−j S. As this has j up movements and (n – j) down movements, the number of paths is given by n the binomial coefficient . To compute the total pseudo-probability of reaching (j) j n−j U D S, multiply the number of ways to reach this node by the probability of each path occurring 𝜋j (1 − 𝜋)n−j . The product is given by n! 𝜋j (1 − 𝜋)n − j ( j! (n − j) ! )

(18.8)

Next, we use this to dress up our two-period model in this binomial garb.

Recasting the Two-Period Example in the Multiperiod Framework In the two-period binomial tree (n = 2), the stock price reaches U2 S or UDS or D2 S at maturity. The topmost node U2 S is reached by two ups in the stock price. Using n = 2 and j = 2, we write the pseudo-probability of the outcome as

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

2 2 2! 𝜋 (1 − 𝜋)0 = 𝜋2 = 𝜋2 (2) 2! (2 − 2)!

(18.9a)

(Note that [1 – 𝜋]0 = 1.) One reaches UDS in two ways: “up and then down” and “down and then up.” Using n = 2 and j = 1, the total pseudo-probability of reaching UDS is 2 1 2! 𝜋 (1 − 𝜋) = 2𝜋 (1 − 𝜋) 𝜋 (1 − 𝜋)2 − 1 = (1) 1! (2 − 2)!

(18.9b)

Two downs in a two-period tree is reached by a unique path. Using n = 2 and j = 0, the total pseudo-probability of reaching D2 S is 2 0 2! 𝜋 (1 − 𝜋)2 = (1 − 𝜋)2 = (1 − 𝜋)2 (0) ( 0! (2 − 0)! )

(18.9c)

One can compactly express expressions (18.9a) – (18.9c) as

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2 j 2! 𝜋 (1 − 𝜋)2 − j = 𝜋j (1 − 𝜋)2 − j (j) ( j! (2 − j)! )

(18.9d)

where j = 2, 1, and 0, respectively These match the pseudo-probabilities determined earlier (for convenience, we also report these values in Figure 18.4). Recall that our two-period call valuation formula (18.6) was 𝜋2 cUU + 2𝜋 (1 − 𝜋) cUD + (1 − 𝜋)2 cDD c= (1 + R)2 Using expression (18.9a), rewrite the first expression in the numerator as 𝜋2 cUU =

2 𝜋2 (1 − 𝜋)0 × max [U2 S − K, 0] (2)

Notice that one can write the call payoff as max[0, U j D2 − j S − K] with j = 2 in this particular instance. Similarly, rewriting the other two terms in the numerator and using the summation notation yields a compact expression for the call’s price: 2

c=

2 j 1 𝜋 (1 − 𝜋)2 − j × max [0, U j D2 − j S − K] ( ) j (1 + R)2 ∑ j=0

(18.10)

THE MULTIPERIOD BINOMIAL OPTION PRICING MODEL

FIGURE 18.4: Two-Period Binomial Model for Pricing European Options Stock t=0

Stock t=1

Stock t=2 U2S

π

Call

cUU = max (U2S – K, 0)

US UDS cUD = max (UDS – K, 0)

S (1 – π)

Pseudo Probability

2 2 π (1 – π)0 = π2 2 2 1 π (1 – π)2 – 1 = 2π(1 – π) 1

DS D2S

cDD = max (D2S – K, 0)

2 0 π (1 – π)2 – 0 = (1 – π)2 0

Money Market Account (mma) Date 0 1

Date 1

Date 2

1+R

(1 + R)2

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The n-Period Binomial Option Pricing Model Replace 2 by n in expression (18.10) to get the general multiperiod binomial model, which is the main result of this chapter.

RESULT 18.2 The n-Period Binomial Option Pricing Model n

c=

n j 1 𝜋 (1 − 𝜋)n − j × max [0, U j Dn − j S − K] ( ) j (1 + R)n ∑ j=0

(18.11)

where n is the number of time periods, T is the option’s maturity date, Δt ≡ T/n is the length of each time period, 1+R ≡ erΔt is the dollar return from the money market account for a time period, j is the numbers of up movements n in n periods (thus [n – j] is the number of down movements), = (j) n! is the number of ways of choosing j “ups” out of n movements, 𝜋 j! (n − j)!

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is the pseudo-probability for the up movement ([1 – 𝜋] is the same for the down movement), U is the up factor for the stock price movement and D is the down factor, S is the initial stock price, and K is the strike price. Replacing the call payoff by max(0, K − Uj Dn−j S) gives the formula for pricing a European put option.

Who but its creator would appreciate such a hideous-looking construct? In its defense, expression (18.11): (1) is a generalization of the simpler formulas introduced earlier, (2) it just uses high school algebra (of course, with some notations), and (3) it is easily programmed into a spreadsheet program such as Microsoft Excel (see the example at the chapter’s end). To reiterate, expression (18.11) generalizes our simple formula for European call pricing developed in the previous section. The first expression within parentheses uses dollar returns to discount the cash flows. Next comes the summation sign, which adds up the “option payoffs” multiplied by the “pseudo-probability of reaching those payoffs” for all possible cases. The result is the option’s price! As in the last chapter, we can rewrite Result 18.2 more abstractly to get an expression similar to the one used in the context of the BSM model (which is explained in the next chapter):

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c = E𝜋 [c (T)] e– r T

(8.12)

where E𝜋 (⋅) denotes computing the expectation under the pseudo-probability 𝜋. Note, as emphasized before, that the option price does not depend on the actual probabilities of the stock price evolution but only on the pseudo-probabilities. Extension 18.1 shows how to choose the up and down factors to obtain a model that approximates the BSM model when the time steps are small. In the limit, these approximations generate a lognormal distribution that underlies the BSM model. You can use Result 18.2 for the one-step valuation of European options in multiperiod binomial trees. The next example demonstrates how to do this when the model has four periods.

EXAMPLE 18.6: A Four-Period Binomial Example ■

This four-period example uses the same BOP data. Given the number of time periods n is 4, divide the time to maturity T into four periods of length Δt = T/n = 0.25 years each.

■

Using specification 1 in Table 18.1, we get U = 1.08458539, D = 0.94056717, 𝜋 = 0.50001507, and (1 – 𝜋) = 0.49998493. Moreover, 1 + R = erΔt = e0.05 × 0.25 = $1.0126 after one period and (1 + R)4 = 1.01264 = $1.0513 after four periods.

THE MULTIPERIOD BINOMIAL OPTION PRICING MODEL

■

407

Using expression (18.11), we get today’s European call price: ⎧ ⎫ ⎪ 4 ⎪ 4 1 4 0 3 1 4 0 3 1 ⎨ ⎬ 𝜋 (1 − 𝜋) max 0, U D S − K + 𝜋 (1 − 𝜋) max 0, U D S − K ( ) ( ) (3) (4) (1 + R)4 ⎪ ⎪ ⎩ ⎭ 1 4 4 =( 0.5000 × 28.3742 + 4 × 0.5000 × 10) 1.0513 ) ( = 4.0653 or $4.07

c=

What happens to the option values if we keep on increasing the number of periods in our numerical example? This is explored in Table 18.1, where we generate options prices by applying specification 1 of Extension 18.1 using finer and finer time intervals. Here Table 18.1a contains some of the Excel workings (for the binomial model for n = 1, 2, 4, 8, 16, and 32; the second panel is a continuation of the computations in the first panel), and Table 18.1b reports call and put prices

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EXTENSION 18.1: Linking the Binomial Model to the Black–Scholes–Merton Model The binomial model is easy to understand and simple to implement, but will it give prices that one can comfortably quote in real markets? The answer is yes! The binomial OPM can be used to approximate the BSM model. The approximation works through approximating the probability distribution for the underlying asset’s price, which the BSM model assumes is a lognormal distribution. The distribution for the binomial process quickly approaches this when the parameter values are properly specified and the model is run over many small intervals. Consequently, options values from the multiperiod binomial models converge to the BSM model. There are several ways of achieving this convergence. “The finance literature has revealed no fewer than 11 alternative versions of the binomial option pricing model for options on lognormally distributed assets,” wrote Chance (2008, p. 38), who noted that the three most widely used versions are Cox et al. (1979), Rendleman and Bartter (1979), and Jarrow and Rudd (1982). The last specification, known as the Jarrow–Rudd binomial model, was developed by Jarrow and Rudd (1982). It builds on Rendleman and Bartter’s approach and was further modified in Jarrow and Turnbull’s (1996) book. In their specification, as the time interval diminishes, the pseudo-probability approaches 0.50 (see Ext. 18.1 Tab. 1). Another specification is the Cox–Ross–Rubinstein binomial model. Though widely used in practice, it gives erroneous results for large time intervals or small volatilities (see Ext. 18.1 Tab. 1). For both specifications, the greater the number of intervals, the better will be the approximation. One must use at least fifty periods to generate good options prices in the binomial models. Which specification is superior? The answer is, they are all equivalent, because they all approach the same BSM model prices as n goes to infinity. Chance (2008, p. 54) concludes that the (binomial) “model is not a single model but a family of interpretations of a discrete-time process that converges to the continuous Brownian motion process [which underlies the BSM models] in the limit and accurately prices options.”

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

Ext. 18.1 Tab. 18.1: Setting Up the Factors and the Pseudo-Probabilities to Link the Binomial and BSM Models Specification 1 (Jarrow–Rudd) U=e

Up factor (U)

rΔt −

𝜎2 2

Specification 2 (Cox–Ross–Rubinstein)

Δt + 𝜎√Δt

e𝜎√Δt

2

rΔt −

Down factor (D) Pseudo-probability of stock going up (𝜋)

𝜎

Δt − 𝜎√Δt

2 D=e (1 + R) − D U−D

1/U (1 + R) − D U−D

U − (1 + R) U−D

U − (1 + R) U−D

Pseudo-probability of stock going down (1 – 𝜋)

rΔt

where Δt is the length of the time interval, r is the continuously compounded annual risk-free interest rate, 1+R = exp(rΔt) ≡ e , and 𝜎2 is the variance of the continuously compounded return.

TABLE 18.1: Convergence of Binomial Option Prices to the Black–Scholes–Merton Model Values

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a. Excel Workings (Binomial Model for n = 1, 2, 4, 8, 16, and 32) S

K

T

n

∆t = T/n

r

1+R

Σ

$100

$110

1

1

1

0.05

$1.0512711

0.14247

100

110

1

2

0.5

0.05

1.02531512

0.14247

100

110

1

4

0.25

0.05

1.01257845

0.14247

100

110

1

8

0.125

0.05

1.00626957

0.14247

100

110

1

16

0.0625

0.05

1.00312989

0.14247

100

110

1

32

0.03125

0.05

1.00156372

0.14247

U

D

𝜋

(1 – 𝜋)

Call

Put

1.199999511

0.90247088

0.500120656

0.499879344

$4.75727159

$9.39250829

1.128249231

0.92236346

0.500042629

0.499957371

4.113492

8.7487287

1.084585393

0.94056717

0.500015067

0.499984933

4.065304709

8.7005414

1.056912736

0.95562533

0.500005326

0.499994674

3.944816311

8.58005301

1.038843626

0.96741588

0.500001883

0.499998117

3.813430739

8.44866743

1.026783114

0.97634426

0.500000666

0.499999334

3.767810681

8.40304738

EXTENDING THE BINOMIAL MODEL

b. Options Prices Number of Intervals

Call Option Price

Put Option Price

1

$4.757272

$9.392508

2

4.113492

8.748729

4

4.065305

8.700541

8

3.944816

8.580053

16

3.813431

8.448667

32

3.767811

8.403047

64

3.803293

8.438529

BSM model

3.783772

8.419009

for these periods (plus that for n = 64). The resulting values converge to the BSM model prices; however, you will have to wait until the next chapter to learn how the BSM model values are obtained.

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18.5

Extending the Binomial Model

The multiperiod binomial model is an all-purpose tool useful for pricing derivatives more complex than the options studied to this point. As such, it is useful to think of the multiperiod model as a computational procedure for valuing arbitrary derivative securities. We illustrate two extensions: (1) including dividends and (2) valuing American options.

Known Dividends US options exchanges do not alter the strike price or make adjustments for cash dividends. Such dividends lower the value of a call but raise the value of a put. Pricing options in the presence of dividends is tricky business. We consider two relatively straightforward ways of adjusting the binomial model for known dividends. As in chapters 3 and 11, the ex-dividend date is the cutoff date for a dividend payment, and we assume that arbitrage ensures that the ex-dividend stock price falls by the dividend (see Result 3.1). Fixed-percentage dividends are easier to model in the binomial framework because the tree still recombines. Unfortunately, fixed-dollar (not fixed-percentage) dividends are the norm among US corporations. Corporations try to hold steady both the dividend date and the dollar amount. Example 18.7 shows how to adjust the model for both fixed-dollar and fixed-percentage dividends.

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EXAMPLE 18.7: Adjusting for Known Dividends (Based on Example 18.2)

Known Dividend Date and a Percentage Dividend Rate ■

In Example 18.2, assume that the stock pays a dividend of 𝛿 = 5 percent just before six months. Then the stock price at time 1 needs to be adjusted to reflect this dividend. Accordingly, the revised stockprices at time 1 are US (1–dividend rate) = 112.824923 (1– 0.05) = 107.18 DS (1–dividend rate) = 92.2363 (1– 0.05) = 87.62

■

Proceed as before. Use these revised stock prices to create the binomial tree. Six-month as well as subsequent stock prices are multiplied by (1 – dividend rate). The tree recombines as before, and our option valuation technique works. Verify that the call value is $2.60 and the put value is $12.23.

Known Dividend Date and a Dollar Dividend

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■

Suppose that in Example 18.2, the stock pays the dividend div = $5 just before six months. In that case, compute the stock price on date 1 as US or DS as before, and then subtract $5 from it. Now create the binomial tree with these revised prices and solve for the options values as before (see Figure 18.5). Note that the tree does not recombine. There are four nodes after one year in a two-period tree (had this been a 10-period tree, you would have 1,024 final nodes!). Solve for the options values by working backward through time, and verify that the call value is $2.77 and the put value is $12.28.

Valuing American Options American options can be easily valued in a binomial model using the following steps: ■

Create a binomial stock price tree as for the European option and find the option values after n periods when the option matures.

■

Using the risk-neutral valuation formula, compute (1) the European option’s value at the “last but one” period (i. e., [n – 1] periods) and (2) the option’s payoff if it is immediately exercised. Retain the larger of these two as the American option’s value and use it in future computations.

■

Work backward period by period. At each node, determine whether the option is worth more alive (computed value) or dead (exercised value), and retain the larger of these two values. The procedure ends when you reach today’s date. The option’s value at time 0 gives the solution. By following this algorithm, you have created the American option price tree.

■

Note that a by-product of the valuation is a description of whether to exercise or not at every node in the tree. Hence the valuation technique also provides the optimal exercise strategy.

EXTENDING THE BINOMIAL MODEL

FIGURE 18.5: Two-Period Tree for Pricing Options When the Underlying Pays a Known Dividend Stock Today

Stock 6 months

Stock 1 year

Call 1 year

Put 1 year

121.65

11.65

0

99.45 98.42

0 0

10.55 11.58

80.46

0

29.54

Dividend paid (112.82 – div) 107.82 100 (92.24 – 5) 87.24

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■

Result 16.10 of Chapter 16 tells us that an American call on a stock that pays no dividends over the option’s life is never exercised early: you get more by selling the call rather than exercising it. Such an American call is equal to a European call, and they have identical binomial trees. In contrast, American put options may be exercised early, and they differ from European puts. Next, we use Examples 18.1–18.3 to create an American put option tree in Example 18.8.

EXAMPLE 18.8: American Option Pricing (Based on Examples 18.1–18.3) ■

Remember that a call option on a stock that pays no dividends over the option’s life is never exercised early. Consequently, the European and American call option trees are the same as in Example 18.2. You can verify this by performing the calculations based on the procedure stated earlier.

■

The story is different for American put options. For convenience, the first tree in Figure 18.6 gives the European put. The second tree shows the American put.

■

Six months before maturity, the European put’s value in the up state is 2.8942. If the put is exercised, the payoff is 110 – 112.8249 = –2.8937 The put is worth more alive than dead—don’t exercise. Retain the value 2.8937 in the American put option tree.

■

Six months before maturity, the put option’s value in the down state is 15.0477. If the put is exercised, the payoff is max (110 – 92.2363, 0) = 17.7637 Here the put is worth more dead than alive. Discard the existing value 15.0494 and replace it with 17.7637 in the tree.

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■

Move back to the present. For the American put, if the put is exercised, it gives max (110 – 100, 0) = 10 Using the American put values from time 1, we find at time 0 P=

𝜋PU + (1 − 𝜋) PD 0.50 × 2.8937 + 0.50 × 17.7637 = = $10.0730 1+R 1.0253

If we exercise put today, we get 110 − 100 = $10. As $10.0730 is the highest, it is recorded as today’s American put value, and the put is not exercised today.

FIGURE 18.6: Two-period Binomial Model for Pricing American Options Put Today

Put 6 months

Put 1 year

European put

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0 2.8942 5.9345

8.7508 15.0494

24.9246

American put 0 2.8942 5.9345

10.0733 15.0494 17.7637

24.9246

SPREADSHEET APPLICATIONS

18.6

Spreadsheet Applications

The binomial model can be implemented using spreadsheet programs like Microsoft Excel. This section demonstrates how to do this in two steps: First, we show how to solve a two-period binomial model, and second, we show how to solve a sixteenperiod example.

A Two-Period Binomial Example You can program the multiperiod binomial OPM in a Microsoft Excel workbook.

EXAMPLE 18.9: Solving Example 18.3 by Excel

Generating Inputs

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■

In an MS Excel worksheet, type in the following. This example uses the BOP data provided earlier in the chapter and used in Examples 18.2 and 18.3. See Figure 18.7 for the worksheet: -

■

S in cell A1 and 100 in cell A2 K in B1 and 110 in B2 T in C1 and 1 in C2 n in D1 and 2 in D2 “Δt = T/n” in E1 and 0.5 in E2 (you can create Δ and other Greek symbols). r in F1 and 0.05 in cell F2 1 + R in G1 and “=EXP(F2*E2)” in G2 and then hit return to get 1.025315. 𝜎 in A4 and 0.142470 in A5

Choose the up and down factors according to one of the specifications connecting the binomial model with the BSM model (see Extension 18.1). We use the Jarrow–Rudd specification: - Type U in B4. For the up factor U = exp[(r – 𝜎2 /2) Δt + 𝜎 √Δt], type “=EXP(((F2– (A5*A5/2))*E2)+(A5*SQRT(E2)))” in B5 and hit return to get 1.128249. - Type D in C4. For the down factor D = exp[(r – 𝜎2 /2) Δt – 𝜎√Δt], type “=EXP(((F2– (A5*A5/2))*E2)–(A5*SQRT(E2)))” in C5 and hit return to get 0.922363.

■

Type 𝜋 in D4 and then “=(G2–C5)/(B5–C5)” in D5 and hit return to get 0.500043. Type (1 – 𝜋) in E4 and then “=1–D5” in E5 and hit return to get 0.499957.

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

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FIGURE 18.7: Two-Period Binomial Example with Excel (Example 18.3)

Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

Column A S

B K

C T

D n

E Δt = T/n

F r

G 1+R

100

110

1

2

0.5

0.05

1.025315

σ 0.142470

U 1.128249

D 0.922363

π 0.500043

(1 – π) 0.499957

Stock (t = 0) Stock (t = 1)

Stock (t = 2) 127.294633

Call (t = 2) Put (t = 2) Pseudo Probability 17.294633 0.000000 0.250043

112.824923 100

104.065586

0.000000

5.934414

85.075435

0.000000

24.924565 0.249957

0.500000

92.236346

EUROPEAN OPTION TREES (METHOD 2) Call tree Call (t = 0)

Call (t = 1)

Call (t = 2) 17.294633

8.434532 0.000000

4.113492 0.000000

0.000000 Put tree Put (t = 0)

Put (t = 1)

Put (t = 2) 0

2.893700 5.934414

8.748729 15.047745

24.924565

EUROPEAN OPTION PRICES (METHOD 3) call = 4.113492 put = 8.748729

SPREADSHEET APPLICATIONS

Creating Binomial Stock Tree and Computing Options Prices and Pseudoprobabilities at Maturity ■

To create the stock tree, type the headings “Stock (t = 0)” in A8, “Stock (t = 1)” in B8, and “Stock (t = 2)” in C8 and then type the following expressions after “=” in each cell; follow up by hitting the return key at the end: -

■

type “=A2” in cell A11 and hit return to get 100 A11*B5 in B10 to get 112.824923 A11*C5 in B12 to get 92.236346 B10*B5 in C9 to get 127.294633 B10*C5 in C11 to get 104.065586 B12*C5 in C13 to get 85.075435

For call prices at maturity, type the heading “Call (t = 2)” in D8 and then type the following after “=” in each cell and hit return at the end: - MAX(C9–B2, 0) in D9 to get 17.294633 - MAX(C11–B2, 0) in D11 to get 0 - MAX(C13–B2, 0) in D13 to get 0

■

For put prices at maturity, type in the label “Put (t = 2)” in E8 and then type the following after “=” in each cell and hit return at the end: - MAX(0, B2–C9) in E9 to get 0 - MAX(0, B2–C11) in E11 to get 5.934414 - MAX(0, B2–C13) in E13 to get 24.924565

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■

To create the pseudo-probabilities of reaching these option values at maturity, type in the label “Pseudo-probability” in F8 and then type and hit return: - D5*D5 in F9 and hit return to get 0.2500043 - 2*D5*E5 in F11 to get 0.500000 - E5*E5 in F13 to get 0.249957 Notice that these pseudo-probabilities sum to 1.

European Call and Put Valuation (Method 2) ■

Type “EUROPEAN OPTION TREES (METHOD 2)” in A16, “Call Tree” in A18

■

You can create the call tree through a repeated application of expression (18.4) of Result 18.1: - Type in the headings “Call (t = 0)” in A19, “Call (t = 1)” in B19, and “Call (t = 2)” in C19. - Copy call values at date 2—type D9 in C20, D11 in C22, and D13 in C24 to get cUU = 17.294633, cUD = 0, and cDD = 0, respectively. - To solve for call values at date 1, type “=(D5* C20 + E5* C22)/G2” in B21 and “=(D5* C22+E5* C24)/ G2” in B23. - Solve and plug in “=(D5* B21+E5* B23)/G2” in cell A22 to get today’s call value 4.113492 or $4.11.

■

This completes the “call tree” that has all possible call values in the binomial tree. The two-period “put tree” is similarly created below the call tree and gives a put value of 8.748729 or $8.75.

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European Call and Put Valuation (Method 3) ■

Type “EUROPEAN OPTION PRICES (METHOD 3)” in A35, “call =” in A36, and “put =” in A37. Using (18.6), we type “= (F9*D9+F11*D11+F13*D13)/(G2ˆ2)” in B36 to get today’s call value 4.113492 or $4.11.

■

Using a variant of (18.6), where calls are substituted with put values, we type “=(F9*E9+F11*E11+F13*E13)/(G2ˆ2)” in cell B37 to get today’s put value 8.748729 or $8.75.

A Sixteen-Period Example Next we extend the preceding example to build a sixteen-period model using the same BOP data.

EXAMPLE 18.10: A Sixteen-Period Example (Excel Application)

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■

We use the BOP data: S is $100, K is $110, T is 1 year, r is 5 percent per year, and 𝜎 is 0.142470. This time, we slice up T into sixteen periods of length Δt each so that Δt = T/n = 1/16 = 0.0625 years. Table 18.3a reports the values obtained by using specification 1 ( Jarrow–Rudd) of Table 18.1: U = 1.038843626 D = 0.96741588 𝜋 = 0.500001883 (1 – 𝜋) = 0.499998117

■

The detailed workings are reported in Table 18.3b. - Column A has j, the number of up movements. - Column B has the call symbols (e. g., in the row corresponding to j = 14, we indicate the call symbol by U14D2S-K; this has stock price for fourteen ups and two downs, from which is subtracted the strike price). - Column C has the call values at expiration. 16 - Column D has the combination, the number of ways of getting j up movements . (j) - Column E has the pseudo-probability of the up movements 𝜋j . - Column F has the pseudo-probability of the down movements (1 − 𝜋)16−j . - Column G has the pseudo-probability of the particular path D × E × F. - Column H has the “expected payoff ” C × G.

SPREADSHEET APPLICATIONS

417

Summing all the values in column H, discounted by (1 + R)16 , gives the call option value, c = $3.81. As we have done the hard work for the call, we can use put–call parity to obtain the put premium: p = c + KB − S = 3.81 + (110/1.0513) − 100 = $8.45

■

TABLE 18.2: A Sixteen-Period Binomial Example (Solved Using Microsoft Excel) a. Basic Data, Computations, and European Call and Put Values S

K

T

n

∆t = T/n

100

110

1

16

0.0625

r 0.05

𝜋

(1 – 𝜋)

Call

Put

0.500002

0.499998

$3.813432

$8.448667

1+R 1.00313

𝜎

U

0.14247

1.038844

D 0.967416

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b. Excel Workings A

B

C

D

E

F

G

H

j

Call Symbol

Call Values at Expiration

Combination

PseudoProbability of Up Jumps 𝜋j

PseudoProbability of Down Jumps (1 – 𝜋)(16–j)

PseudoProbability of Particular Path D*E*F

Expected Payoff C*G

16

U16S–K

73.993666

1

0.000015

1.000000

0.000015

0.001129

15

U15DS–K

61.342818

16

0.000031

0.499998

0.000244

0.014977

14

U14D2S–K

49.561804

120

0.000061

0.249998

0.001831

0.090754

13

U13D3S–K

38.590817

560

0.000122

0.124998

0.008545

0.329768

12

U12D4S–K

28.374162

1,820

0.000244

0.062499

0.027772

0.788002

11

U11D5S–K

18.859973

4,368

0.000488

0.031249

0.066652

1.257053

10

U10D6S–K

9.999951

8,008

0.000977

0.015625

0.122194

1.221936

9

U9D7S–K

1.749117

1,1440

0.001953

0.007812

0.174562

0.305329

Sum/1.051271 = 3.813431

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

18.7

Summary

1. The basic binomial OPM is described by the following: a. Three securities trade: a stock, a riskless money market account, and a European (call or put) option. b. The stock price goes up by factors of U or D from each node. The stock price tree repeats this single-period structure at the end of each node. c. A dollar invested in the money market account gives a dollar return of 1 + R ≡ erΔt after one period. d. Given the strike price, you can compute the option’s payoffs at maturity from the stock tree. Then solve the previous period’s option price by repeatedly applying the formula Option price = [𝜋 × (Option price)UP + (1 – 𝜋) × (Option price)DOWN] / (1 + R) where 𝜋 = [(1 + R) – D]/(U – D) is the pseudo-probability of an up movement and the options prices “up” and “down” are considered at the next up and down nodes. 2. The n-period multiplicative binomial OPM is n

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n j 1 c= 𝜋 (1 − 𝜋)n − j × [max (0, Uj Dn−j S − K)] n ∑ } {[( ] j) (1 + R) j=o n ≡ n! / [j!(n – j)!] is the number of ways of choosing j “ups” out of (j) n movements and n! = n × (n – 1) × (n – 2) × … × 2 × 1 is the n-factorial.

where

3. An American option may be valued in the binomial framework as follows: a. Create a binomial stock price tree as in the case of the European option and find the option values at time n, which is the maturity date. b. Using the risk-neutral valuation formula, compute (1) the European option’s value at time (n – 1) and (2) the option’s payoff if it is immediately exercised. Retain the larger of these two as the American option value and use it in future computations. c. Work backward period by period. At each node, determine whether the option is worth more alive (computed value) or dead (exercised value), and retain the higher of these two values. Stop when you reach today’s date. By following this process, you have created the American option price tree.

QUESTIONS AND PROBLEMS

18.8

Cases

Leland O’Brien Rubinstein Associates Inc.: Portfolio Insurance (Harvard Busi-

ness School Case 294061-PDF-ENG). The case studies the rise and fall of Leland O’Brien Rubinstein (LOR) Associates’ portfolio insurance selling business. Leland O’Brien Rubinstein Associates Inc.: SuperTrust (Harvard Business

School Case 294050-PDF-ENG). The case examines LOR Associates’ attempts to rebuild itself after the 1987 stock market crash by creating new products to meet the unsatisfied needs of equity investors. The Value of Flexibility at Global Airlines: Real Options for EDW and CRM

(Kellogg School of Management Case, Case KEL266-PDF-ENG, Harvard Business Publishing). The case evaluates real options for technology projects using a binomial model.

18.9

Questions and Problems

18.1. Why is a multiperiod binomial model a better approximation to the actual

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stock price process than the single period binomial model? The next seven questions are based on the following data for a two-period binomial model - A stock’s price S is $100. After three months, it either goes up and gets multiplied by the up factor U = 1.163287, or it goes down and gets multiplied by the down factor D = 0.861785. - Options mature after T = 0.5 years and have a strike price K = $110. - A dollar invested in the money market account grows to 1 + R = $1.012578 after three months. The stock price tree is: Stock Today

Stock 3 months

Stock 6 months

135.3238 116.3287 100.00

100.2503 86.1785 7 4.2673

18.2. Compute the call value in the above tree using synthetic construction. 18.3. Demonstrate how you can make arbitrage profits when a trader quotes a call

price of $2.

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CHAPTER 18: MULTIPERIOD BINOMIAL MODEL

18.4. Compute the value of the call option using risk neutral valuation. 18.5. Compute. the put value in the above tree using synthetic construction. 18.6. Suppose the market price of the put option in question 18.5 is $10, how

would you take advantage of this mispricing? Explain your answer. 18.7. Given the data in question 18.5, compute the value of the put option using

risk neutral valuation. 18.8. Consider the call and put options in questions 18.2 and 18.5. Show that they

satisfy put–call parity. The next five questions are based on the following data for a four period binomial model - Options mature after T = 2 years and have a strike price K = $75. - The price of the underlying stock price is $50. The stock price evolves according to the Jarrow–Rudd specification in this 4-period model with volatility 𝜎 = 0.3. - The continuously compounded risk-free interest rate r is 5 percent per year. 18.9. a Compute the up and down factors for the stock price movements and the

dollar return (1 + R) for each period. b Use these up and down factors to create a 4-period tree for the stock price

movements. 18.10. Find the value of all call options in the tree by repeated application of risk

neutral valuation. 18.11. Explain the European call option formula at time 0 based on an n-period Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

binomial option pricing model. 18.12. Find the value of all put options in the tree by repeated application of

riskneutral valuation. 18.13. Explain the European put option formula at time 0 based on an n-period

binomial option pricing model. The next eight questions are based on the following data for a two-period binomial model - Options mature after T = 0.5 year and have strike price K = $45. - The price of the underlying stock price is $50. The stock price evolves according to the Jarrow–Rudd specification in this 2-period model with volatility 𝜎 = 0.2. - The continuously compounded risk-free interest rate r is 2 percent per year. 18.14. Compute the up and down factors, the dollar return (1 + R), and the stock

price lattice. 18.15. Compute today’s call option price in this 2-period tree. 18.16. Compute today’s put option price in this 2-period tree. 18.17. Consider an otherwise identical American call option with the preceding data,

maturing at time 2. Compute its value. How do the American and European call prices compare?

QUESTIONS AND PROBLEMS

18.18. Consider an otherwise identical American put option with the preceding data,

maturing at time 2. Compute its value. How do the American and European put prices compare? 18.19. Compute today’s American put option price in this 2-period tree if the strike

price is $70. Does early exercise occur? 18.20. Consider the following exotic option whose payoff at maturity is given by the

stock price squared less a strike price if it has a positive value, zero otherwise: max [S(2)2 – K, 0] . Assuming that the strike price is $2,000, determine the value of this exotic option under the assumption of no-arbitrage. 18.21. Consider the following exotic option whose payoff at maturity is given by

the square root of the stock price less the strike price if it has a positive value, zero otherwise: max [√S (2) – K, 0] . Using the preceding data except for assuming a new strike price is $5, determine the value of this exotic option under the assumption of noarbitrage. (Microsoft Excel) Consider the following

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- Options mature after T = 0.5 year and have strike price K = $65. - The price of the underlying stock S is $63. The stock price evolves according to the Jarrow–Rudd specification in this 3-period model with volatility 𝜎 = 0.3. - The continuously compounded risk-free interest rate r is 0.045 per year. 18.22. a. Using Excel, compute today’s call option value with the preceding data. b. Using Excel, compute today’s put option value with the preceding data.

421

19 The Black–Scholes–Merton Model 19.1 Introduction 19.2 Nobel Prize–Winning Works (1973)

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19.3 The Assumptions EXTENSION 19.1 Bubbles and Option Pricing

19.4 The Pricing and Hedging Argument 19.5 The Black–Scholes–Merton Formula 19.6 Understanding the Black–Scholes–Merton Model Stock Prices and Martingales Risk-Neutral Valuation Actual versus Pseudo-probabilities

19.7 The Greeks Interpreting the Greeks Some Road Bumps Ahead EXTENSION 19.2 Market Manipulation and Option Pricing

19.8 The Inputs Observable Inputs Volatility: The Elusive Input

19.9 Extending the Black–Scholes–Merton Model Adjusting for Dividends Foreign Currency Options Valuing American Options EXTENSION 19.3 Exotic Options

19.10 Summary 19.11 Appendix Modeling the Stock Price Evolution Continuously Compounded (Logarithmic) Return Introducing Uncertainty The Central Limit Theorem First Derivation of the Black–Scholes–Merton Formula (as a Limit of the Binomial Model) Second Derivation of the Black–Scholes–Merton Formula (Using Risk-Neutral Valuation) The Probabilities and the Risk Premium Proof of (24)

19.12 Cases 19.13 Questions and Problems

NOBEL PRIZE–WINNING WORKS (1973)

19.1

Introduction

The year 1973 saw the publication of the Black–Scholes–Merton (BSM) option pricing model (OPM) and the birth of the first modern options exchange, the Chicago Board Options Exchange. Professor Mark Rubinstein declared in his 1994 American Finance Association presidential address, “This model [BSM model] is widely viewed as one of the most successful in the social sciences and has perhaps been (including its binomial extension) the most widely used formula, with embedded probabilities, in human history.” We share the professor’s enthusiasm, but we are a little less sure about the usage data.1 This chapter presents the BSM model. First, we consider some applications and uses. Next, we provide a brief history of the events that led to its development and publication. We describe the model’s assumptions, present the Black–Scholes formula for valuing European options, and discuss the theory underlying the model’s structures. We discuss how to gather the inputs, including the most elusive, the “volatility.” Finally, we show how to extend the BSM model to adjust for dividends, to price foreign currency options, and we discuss the pricing of American options. An insert presents a bird’s-eye view of exotic options.

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19.2

Nobel Prize–Winning Works (1973)

During the late 1960s, a Harvard-trained applied mathematician, Fischer Black, was working as a consultant in Boston. Black applied the capital asset pricing model to value options and derived a partial differential equation (pde) that implicitly solved the option’s value. He teamed up with Massachusetts Institute of Technology (MIT) assistant professor Myron Scholes to solve the equation and derive the formula that bears their names. At the same time, a doctoral student of Paul Samuelson, Robert Merton, was pioneering the use of stochastic calculus to tackle a variety of problems, including portfolio theory and option valuation. Black and Scholes solved the pde using the Capital Asset Pricing model. Later, Merton showed Black and Scholes how to derive their pde using a perfectly hedged portfolio consisting of the stock and the option. The Black–Scholes paper had difficulty getting published, Professors Merton Miller and Eugene Fama used their personal influence to get the paper published in the Journal of Political Economy as “The Pricing of Options and Corporate Liabilities” in 1973. The same year also saw the publication of Merton’s (1973a) “Theory of Rational Option Pricing” in the Bell Journal of Economics and Management Science. The rest is history. The crowning academic glory came in 1997 when Merton and Scholes won the Bank of Sweden Prize in Economic Sciences in Memory of Alfred 1

In George Bernard Shaw’s play “O’Flaherty V. C.,” an English general told an Irish war hero, “Well, in recruiting a man gets carried away. I stretch it a bit occasionally myself. After all, it’s for king and country. But if you won’t mind my saying it, O’Flaherty, I think that story about your fighting the Kaiser and the twelve giants of the Prussian guard singlehanded would be the better for a little toning down. I don’t ask you to drop it, you know; for it’s popular, undoubtedly; but still, the truth is the truth. Don’t you think it would fetch in almost as many recruits if you reduced the number of guardsmen to six?”

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

Nobel. Black, whose contributions were duly acknowledged in the Nobel citation, unfortunately died from cancer two years earlier and missed the honor.

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19.3

The Assumptions

The BSM model, like its binomial approximation, requires several important assumptions. We repeat here assumptions A1–A6 of Chapter 17, which are common to both models and restate much of the discussion. The last assumption (binomial trading process) is replaced by two assumptions (A7 and A8) that characterize the stock price evolution under continuous trading. A1. No market frictions. We assume that trading involves no market frictions such as transaction costs (brokerage commissions and bid/ask spreads), margin requirements, short sale restrictions, and taxes (which may be levied on different securities at different rates). Moreover, assets are perfectly divisible. This helps create a benchmark model to which one can later add frictions, and it’s a reasonable approximation for institutional traders. A2. No credit risk. We assume that there is no credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty will fail to perform on an obligation. The absence of credit risk is a reasonable assumption for exchange traded derivatives. Over-the-counter derivatives, however, may possess substantial credit risks, unless there are adequate collateral provisions. A3. Competitive and well-functioning markets. In a competitive market, traders’ purchases and sales have no impact on the market price; consequently, traders act as price takers. In a well-functioning market, price bubbles are absent. The competitive markets assumption is the workhorse of modern economics. The relaxation of this assumption is discussed in Extension 19.2, after we present the BSM model. A price bubble happens when an asset’s price substantially deviates from its intrinsic or fundamental value. Recent research shows that bubbles can invalidate the methodology we employ for pricing options (see Extension 19.1). A4. No intermediate cash flows. Many assets reward their holders with cash flows such as dividends for stocks and coupons for bonds. The BSM model assumes that the underlying asset has no cash flows over the option’s life. Later in the chapter, we will show how to relax this assumption. A5. No arbitrage opportunities. This is self-explanatory.

EXTENSION 19.1: Bubbles and Option Pricing As discussed in Chapter 17, a price bubble happens when an asset’s price substantially deviates from its intrinsic or fundamental value. The fundamental or intrinsic value can be defined in two equivalent ways: the first is the present value of its future cash flows; the second is the price paid if, after purchase, you have to hold the asset forever. A difference only occurs if one believes that selling can generate a higher value. This difference is the bubble.

THE ASSUMPTIONS

425

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OPMs, like any other models, are valid under a given set of assumptions. When an assumption fails, the model may no longer be valid. If assumption A3 (competitive and well-functioning assumption) fails and a bubble exists, then many results of option pricing theory no longer hold (see Jarrow, Protter, and Shimbo 2010b). Fortunately, with respect to the BSM model, assumption A8 is consistent with assumption A3. This is because the assumption that the stock price follows a lognormal distribution excludes bubbles. Thus, the lognormal distribution is a very strong assumption. If one believes that there are stock price bubbles, then the lognormal distribution assumption needs to be modified, and the BSM model no longer applies. It can be shown that in a market with no arbitrage opportunities or dominated assets, asset price bubbles only exist when the market is incomplete. Because most option pricing theories assume a complete market (where, recall from Chapter 8, securities trade that help investors attain any desired future payoffs), so that synthetic construction is possible, we see that the standard approach to option pricing excludes bubbles. Again, this is a good result, as long as one believes that there are no stock price bubbles. But how do you price options when there are asset price bubbles in an incomplete market? In such markets, alternative pricing methodologies need to be invoked. The two common approaches are using indifference/utility-based pricing and identifying the risk-neutral probabilities from traded derivatives. Because exact hedging is no longer possible, we cannot price by synthetic construction; however, risk-neutral valuation may still apply. When there are bubbles in the underlying asset, pricing is subtle. Recent research suggests that some results stay the same while others change: ■

Put–call parity still holds for European options in the presence of bubbles.

■

Risk-neutral valuation works for the put’s price because it has a bounded payout, but it does not work for the call; it turns out that if the underling asset has a price bubble, so does the call.

■

Forwards and futures both inherit the underlying’s price bubble, and even more interesting, futures prices can have their own price bubbles.

Bubbles are a thorn in the side of the policy makers, a source of problems for asset holders, an opportunity for hedge fund managers, and a hot research area for professors. The three witches in William Shakespeare’s Macbeth sang together, “Double, double, toil and trouble/Fire burn and cauldron bubble.” Perhaps we should rephrase: “Bubble, bubble, toil and trouble!”

A6. No interest rate uncertainty. We assume that interest rates are constant across time. This assumption works for short-lived options whose underlying asset isn’t sensitive to interest rate movements. This assumption is relaxed in part IV of the book. Next, we introduce two assumptions that are crucial for developing the BSM model A7. Trading takes place continuously in time. This assumption gives a realistic description of transacting actively traded stocks. Stocks do not trade only on a fixed time grid but rather at any time instant during the trading day. This is particularly true in today’s markets, where a significant amount of trading volume is attributed to high-frequency computerized trading strategies pursuing fleeting profit opportunities in the blink of an eye.

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

A8. The stock price follows a lognormal probability distribution. This assumption means that the continuously compounded returns on the stock follow a normal distribution. We will explain this assumption in two ways: first, by some diagrams, and second, by adding more structure to the continuously compounded rate of return concept familiar from Chapter 2. Does the lognormal assumption give a realistic model of stock price behavior? As pictures say a thousand words, let’s go back to our diagrams. In Figure 19.1, the first two diagrams repeat possible stock price evolutions from Figure 18.1, for the single period and multiperiod binomial models, respectively. As “seeing is believing,” you can easily surmise, “These don’t look like stock price paths!” The multiperiod binomial evolution diagram leaves open the intriguing possibility that it might start looking like a stock price path if the time step shrinks to zero. This brings us to the third figure, which depicts a stock price path for the lognormal distribution. Here you can no longer rule out the evolution just by visual inspection. A rejection of this evolution now requires the use of formal statistical procedures. Hence it is the first “real” or “realistic” stock price evolution that we have considered. Next, let’s use Result 2.1c (with slightly enhanced notation): an investment of L(0) ≡ L dollars in a risk-free account today grows to L(T) ≡ L(0)erT dollars after T years, where r is the continuously compounded risk-free rate of return per year. Is there a similar result for stocks? Yes. We can write

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S (T) = S (0) eyT

(19.1)

where S(T) and S(0) are the stock prices at times T and 0, respectively, and y is the continuously compounded rate of return on the stock per year. However, this return is not risk-free. To characterize its properties, we need a model of returns that (1) generates prices that look like the stock price patterns observed in financial markets, (2) never allows stock prices to be negative (because they are of limited liability), (3) but allows stock price returns to be positive, zero, or negative (as you find in the markets), and (4) provides a mathematically tractable model. Samuelson, Merton, and other researchers achieved these objectives in one stroke by replacing Bachelier’s arithmetic Brownian motion with a geometric Brownian motion. The modification gives us a structure similar to (19.1). The geometric Brownian motion assumption is that y is normally distributed with mean (𝜇 − 𝜎2 /2)T and standard deviation 𝜎√T. This is expressed as 𝜇T −

S (T) = S (0) e

𝜎2 T 2

+ 𝜎√Tz

(19.2)

where 𝜇 is the expected return on the stock per year, 𝜎 is the standard deviation or volatility of the stock’s return per year, and z is a standard normal random

THE ASSUMPTIONS

variable with mean 0 and variance 1; that is, z has the probability density function (a plot of which gives the familiar bell curve) q (z) =

1

−

√2𝜋

e

z

2

2

(19.3)

FIGURE 19.1: Stock Price Evolution under Discrete and Continuous Trading Single Period Stock price Realized stock price path (single period model) Possible path

US S DS 0

Today

After 1 year Time

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Multiple Periods Stock price Actual stock path Stock price path (multiperiod model)

... US S DS ... 0

Now

Continuous Time Path Stock price

Actual stock path Stock path (lognormal model)

S

0

1 year Time

Now

1 year Time

The probability q(z) is the actual probability of the random variable z, which generates the randomness in the stock price S(T).

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Note that Equation 19.2 satisfies properties 1, 2, and 3 trivially. Property (4) follows because the normal distribution is so well studied. The appendix to this chapter provides an axiomatic justification for modeling the stock price with a lognormal distribution based on the efficient markets hypothesis.

19.4

The Pricing and Hedging Argument

The original proof of the BSM model utilizes the idea of hedging an option position with an opposing stock trade, a notion used by options writers in London more than a century ago (Nelson 1904). An understanding of this hedging argument is essential to understanding the BSM model. The intuition is easily explained in three steps (see Figure 19.2). Step 1 ■

Call prices and the underlying stock price move together. This happens because as the stock price goes up, the call has a higher chance of ending up in-the-money, which makes the call more valuable today.

■

Consequently, the call value is a function of the underlying stock price, and because the option eventually matures, its price must also depend on time.

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Step 2 ■

If a long call and a long stock position move together, then a long call and a short stock position must move in opposite directions. Recall from Chapter 15 that this creates a partial hedge because the increase in one is offset by a decrease in the other.

■

But they don’t perfectly balance, so it is not a perfect hedge.

Step 3 ■

We can modify the partially hedged position in the long call and short stock to make it a perfect hedge. One can do this by short selling a fraction of a stock for each long call. Then, for any movement in the stock price, the change in the call value exactly offsets the change in the short stock position.

■

The perfectly hedged portfolio uniquely identifies the option’s arbitrage-free price. This happens because the hedged portfolio requires a known initial investment, and it is riskless over the next time period. To avoid arbitrage, it must earn the risk-free rate. The equation implied by this restriction, a partial differential equation, when solved yields the option’s price. The mathematics for this argument is contained in the appendix to Chapter 20.

The replication argument as just described utilizes the hedge ratio, which we called the holy grail of option pricing. Jarrow (1999, p. 233) notes that “the idea of constructing a perfectly hedged portfolio is the key insight of the Black–Merton– Scholes model, more important than the valuation formula itself. Indeed, if one considers the meaning of a perfectly hedged portfolio, it becomes apparent that

THE BLACK–SCHOLES–MERTON FORMULA

FIGURE 19.2: The Pricing and Hedging Argument Stock and call prices move together

Call and short stock positions move in opposite directions

Call and a fraction of a short stock creates a perfectly hedged portfolio

A riskless portfolio of the stock and option changing continuosly in time earns the risk-free rate

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Black–Scholes–Merton model This gives a pde whose solution is the BSM model

it implies that a position in the stock and the riskless asset can be created that exactly duplicates the changes in value to the call option” [emphasis added]. The creation of synthetic options is the basis for most OPMs and will be used throughout this book.

19.5

The Black–Scholes–Merton Formula

Under assumptions A1–A8, one can derive the Black–Scholes–Merton formula for pricing European options. There are three ways of proving this formula: (1) taking the limit of the multiperiod binomial model of Chapter 18 as the time steps go to zero, (2) directly applying risk-neutral valuation, and (3) using the hedging argument of the previous section to obtain a partial differential equation (pde) whose solution yields the formula. The appendix to this chapter provides the proof using the multiperiod binomial model, the simplest derivation, and the proof using risk-neutral valuation, the paramount technique of modern derivatives pricing. The appendix to Chapter 20 provides the pde proof, the derivation originally used by Black and Scholes (1973) but the least extendable methodology of the three.

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The formula is given by the following result (see Result 19.1).

RESULT 19.1 The Black–Scholes–Merton Formulas for Pricing European Options The European call price is c = SN (d1 ) − Ke−rT N (d2 )

(19.4a)

where S ≡ S(0) is today’s spot price, N( . ) is the cumulative standard normal distribution function which has mean 0 and standard deviation 1,

d1 =

𝜎2 S T log ( ) + r + ( K 2)

𝜎√T d2 = d1 − 𝜎√T

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K is the strike price, r is the continuously compounded risk-free interest rate per year, T is the time to maturity measured in years (time 0 is today), and 𝜎 is the stock return’s volatility per year, which is the square root of the stock return’s instantaneous variance. The European put price is given by p = Ke−rT N (−d2 ) − SN (−d1 )

(19.4b)

One can show that the put price follows from the call’s price using put–call parity for European options along with the result that N(d) + N(–d) = 1. The first observation to make about the BSM formula is that the call’s value depends on the stock price (S), the strike price (K), the time to maturity (T), the riskless interest rate (r), and the stock return’s volatility (𝜎). It does not depend, surprisingly, on the stock’s expected return (𝜇). This omission is significant and has two important implications. First, the stock’s expected return is the most difficult input to estimate. The estimation of the other inputs is easier and discussed in a subsequent section. In fact, had the option’s value depended on 𝜇, the formula would not have been used. Estimating 𝜇 is equivalent to estimating the stock’s risk premium, and no two financial economists agree on how to do this. This is a well-known problem in portfolio theory. Consequently, avoiding the estimation of 𝜇 is a blessing in disguise. The second implication is that because the option value does not depend on 𝜇, two traders who agree on (S, K, T, r, 𝜎) but disagree on the stock’s expected return, 𝜇, will still agree on the call’s value. Of course, if they disagree on the stock’s expected return, 𝜇, the traders will probably disagree on the stock price S. However, if they agree on S, regardless of their views of 𝜇, they agree on the call’s value.

THE BLACK–SCHOLES–MERTON FORMULA

Example 19.1 demonstrates how to use Result 19.1, assuming that you are given all of the inputs.

EXAMPLE 19.1: Using the Black–Scholes–Merton Formula Consider the binomial option pricing (BOP) data of Chapter 18: ■

YBM’s current stock price S is $100.

■

The strike price K is $110.

■

Time to maturity T is 1 year.

■

The volatility 𝜎 of the stock’s return is 0.142470 per year.

■

The continuously compounded risk-free interest rate r is 5 percent per year.

The European Call Price ■

Compute d1 and d2 : 𝜎2 S T log ( ) + r + ( K 2)

log (

(0.14247)2 100 (1) + 0.05 + ] 110 ) [ 2

= 0.14247√1 𝜎√T d2 = d1 − 𝜎√T = − 0.2468 − 0.14247 = −0.3893

d1 =

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■

= − 0.2468

One can get the cumulative normal distribution function from (1) a spreadsheet program such as Excel, (2) a calculator, or (3) a table of values for the standard normal distribution (Appendix A at the end of the book shows how to read this table)2 . Using Excel, we get N (d1 ) = N (−0.2468) = NORMSDIST (−0.2468) = 0.4025 N (d2 ) = N (−0.3893) = NORMSDIST (−0.3893) = 0.3485

■

The zero-coupon bond price is B = erT = e−0.05 = 0.9513.

■

The European call price using the BSM formula is c = SN (d1 ) − Ke−rT N (d2 ) = 100 × 0.4025 − 110 × 0.9513 × 0.3485 = $3.78

The European Put Price ■

For European puts, we need different values from the cumulative normal distribution function: N (−d1 ) = N [− (−0.2468)] = N (0.2468) = NORMSDIST (0.2468) = 0.5975 N (−d2 ) = NORMSDIST (0.3893) = 0.6515

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■

The European put price is p = Ke–rT N (−d2 ) − SN (−d1 ) = 110 × 0.9513 × 0.6515 − 100 × 0.5975 = $8.42

The European Put Price (by Put–Call Parity) ■

We can double-check our computations using put–call parity: p = c + Ke−rT − S = 3.78 + 110 × 0.9513 − 100 = $8.42

2

In Excel, the command “=NORMSDIST(z)” gives the standard normal cumulative distribution function (in Lotus Notes, use the command @NORMAL[z, 0, 1, 0]). The distribution has a mean of 0 and a standard deviation of 1.

19.6

Understanding the Black–Scholes–Merton Model

To better understand the BSM model, we apply the same argument used to obtain the binomial model in Chapter 17. This argument starts by introducing an analogous notion to that of the pseudo-probability.

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Stock Prices and Martingales In an arbitrage-free market, it can be shown (as in the binomial model) under the lognormal assumption that there exist pseudo-probabilities such that the stock price today is its discounted expected value at time T, that is, S (t) = E𝜋t [S (T)] e−r(T−t)

(19.5)

where 𝜋(z) are the pseudo-probabilities. This is analogous to Equation 17.7. Equation 19.5 is a present value formula. In this formula, the adjustment for risk occurs in the use of the pseudo-probabilities when taking the expectation and not in the discount rate, which is riskless. Rearranging terms in Equation 19.5, we get S (T) S (t) = E𝜋t rt [ e erT ]

(19.6)

where the t corresponds to the date the expectation is taken. In this form, analogous to Chapter 17, we see that the stock price normalized by the price of a money market account is a martingale under the pseudo-probabilities. As noted in Chapter 17, martingales in probability theory are associated with fair games, that is, games in which the gambler has an equal chance of losing or winning. In our context, this is an implication of the no arbitrage assumption. In fact, although not

UNDERSTANDING THE BLACK–SCHOLES–MERTON MODEL

proven here, the implication also goes in the reverse direction. These insights are the basis for many generalizations of the BSM formula. Next, we apply this approach to value a European call option.

Risk-Neutral Valuation Under the lognormal distribution assumption, the market is complete, and we can construct a portfolio in the stock and money market account that exactly replicates the payoffs to the traded call option. As before, the cost of constructing this synthetic option yields the arbitrage-free price of the traded call, and exactly as in Chapter 17, this implies that the call price can be computed as its discounted expected payoff using the pseudo-probabilities and the risk-free rate; that is, risk-neutral valuation applies: c = E𝜋 {max [0, S (T) − K]} e−rT

(19.7a)

It is shown in the appendix to this chapter that one can rewrite this equation in the following way: c = E𝜋 {S (T) |S (T) ≥ K} Prob𝜋 {S (T) ≥ K} e−rT −KProb𝜋 {S (T) ≥ K} e−rT

(19.7b)

Making the identification of the terms in Equation 19.7b with those in the BSM formula (19.4), we see that

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S (0) N (d1 ) = E𝜋 {S (T) |S (T) ≥ K} Prob𝜋 {S (T) ≥ K} e−rT KN (d2 ) e−rT = K Prob𝜋 {S (T) ≥ K} e−rT

(19.8)

The identifications help explain the meaning of the BSM formula. We now see that the first term in the BSM formula, S(0)N(d1 ), can be interpreted as the discounted expected stock price at maturity, given that it ends up inthe-money multiplied by the probability of being in-the-money. This interpretation makes intuitive sense. Of course, as just noted, the expectation is computed using the pseudo-probabilities to adjust for the risk of the cash flows. We note that because the current stock price S(0) is not the expected stock price at maturity, given that it ends up in-the-money, N(d1 ) is not the probability of the stock being in-the-money. The second term in the BSM formula, KN(d2 )e−rT , can be interpreted as the discounted strike price (Ke−rT ) times the pseudo-probability of ending up in the money, that is, N(d2 ) = Prob𝜋 {S(T) ≥ K}. Again, this interpretation makes intuitive sense. This explains why the hedge ratio N(d2 ) is always between zero and 1, because the probability of being in-the-money is always between 0 and 1. It is 0 only when the stock price will never be in-the-money, and it is 1 only when the stock price will be in-the-money for sure.

Actual versus Pseudo-probabilities Are the actual (q[z]) and pseudo (𝜋[z]) probabilities linked? Interestingly, and not surprisingly, they are linked by a risk premium. The argument is a simple

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generalization of our discussion in Chapter 17. The details are given in the appendix to this chapter. First, note that 𝜋 (z) > 0 if and only if q (z) > 0 (19.9a) that is, the actual and pseudo-probabilities must agree on zero-probability events. This follows from the no arbitrage assumption. Consistent with this, one can show that the pseudo-probabilities equal 𝜋 (z) = 𝜙 (z) q (z)

(19.9b)

2

where 𝜙(z) = e−(𝜃 /2)+𝜃z > 0 and 𝜃 = −[(𝜇 − r)/𝜎]√T. The quantity 𝜙(z) in Equation 19.9b represents an adjustment for risk, and 𝜃 is, in fact, the stock’s risk premium. The appendix to this chapter provides the justification for these statements. Notice that the result looks identical to that from Chapter 17.

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19.7

The Greeks

We can gain a better understanding of the BSM model by understanding how its prices change when we change the inputs. This is easily accomplished by taking partial derivatives (given in Result 19.1) with respect to the stock price (S), time to maturity (T), the riskless interest rate (r), and the stock return’s volatility (𝜎). In addition, we also compute the second partial derivative with respect to the stock price. These partial derivatives are called the Greeks because they are often written using the Greek alphabet—although one of them has a fictitious Greek name! These Greeks are given in Table 19.1. They are also useful for hedging and portfolio risk management, which are discussed in Chapter 20.

Interpreting the Greeks We first consider the delta, which measures the change in the option’s value when there is a small change in the underlying stock price. For the call, we have Delta ≡

Change in the value of the call 𝜕c = = N (d1 ) 𝜕S Change in the value of the stock

(19.10a)

The term on the right side of this equation is defined in Result 19.1. This shows that the delta, the hedge ratio, is a number strictly between zero and one. We can similarly compute the delta for a put. The sensitivity of the delta to changes in the underlying stock price is called the gamma. This is the second partial derivative of the BSM call price formula with

THE GREEKS

respect to the stock price, that is Change in the delta 𝜕 (Delta) 𝜕2 c = = 𝜕S Change in the value of the stock 𝜕S2 1 = N′ (d1 ) > 0 S𝜎√T

gamma ≡

(19.10b)

where N′( . ) is the standard normal density function and all of the remaining symbols are defined in Result 19.1. The gamma shows that as the stock price increases, the delta increases. The gamma for the put is similarly computed by taking the partial derivative with respect to the underlying stock price of the delta for the put. The gamma for the put and the call are identical. Example 19.2 illustrates these computations.

EXAMPLE 19.2: Deltas and Gammas for European Options ■

Consider the BOP data (see Example 19.1): S = $100, K = $110, T = 1 year, 𝜎 = 0.142470 per year, and r = 0.05 per year. Using the formulas from Table 19.1 and computations from the previous example, we get d1 = −0.2468. Then,

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For the call, deltaC = N(d1 ) = 0.4025 For the put, deltaP = −N(−d1 ) = −0.5975 ■

We use Microsoft’s Excel spreadsheet program for computing deltas and plotting their values. The deltas are computed using the NORMSDIST function for stock prices S at $5 intervals ranging from $50 to $150 (see Table 19.2) and are drawn in Figure 19.3a, where the stock price is depicted on the horizontal axis and the delta values are along the vertical axis. Notice that both the deltas are upward sloping curves.

■

To understand these figures, consider the behavior of the option’s price for extreme stock price values. - For stock prices near $150, the call is deeply in-the-money and increases by almost a dollar for a dollar increase in the stock price, which gives a delta close to 1. In contrast, the put is deeply out-of-the-money and has a near-zero response to any stock price movement, signifying that the put’s delta is close to 0. - For very low stock prices, the call’s delta is almost 0, while the put option’s price decreases by almost a dollar for each dollar increase in the stock price, yielding a delta close to −1.

■

The gamma is the same for calls and puts. Using the NORMDIST function in Microsoft Excel’s spreadsheet to compute N′(d1 ), we get gamma = N′(d1 ) / (S𝜎√T) = 0.0272 We compute the gammas for stock prices S = $50, 55, … ,145, 150; report them in Table 19.2; and plot them in Figure 19.3b.

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TABLE 19.1: Partial Derivatives of the Black–Scholes–Merton Model Derivative 𝜕BS Delta ≡ 𝜕S 𝜕2 BS Gamma ≡ 𝜕S2 𝜕BS 𝜕t 𝜕BS Vega ≡ 𝜕𝝈 𝜕BS Rho = 𝜕r Theta ≡

Call Option

Put Option

0 < N(d1 ) < 1

−1< −N(−d1 )< 0

1

N′(d1 ) > 0 S𝜎√T SN′(d1 )𝜎 − − rKe−rT N(d2 ) < 0 2√T

Same as for European call −

SN′(−d1 )𝜎 2√T

+ rKe−rT N(−d2 )

S√TN′(d1 ) > 0

Same as for European call

TKe−rT N(d2 ) ≥ 0

−TKe−rT N(−d2 ) ≤ 0

> 0
0 As the volatility increases, the value of the call increases. The same is true for the put option. Increased volatility is a good thing for the option’s value! The larger the volatility, the more spread out the stock price distribution is and the larger is the tail

THE GREEKS

probability that the option ends up in-the-money. It’s no wonder that most traders view options as “volatility bets”!

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TABLE 19.2: Deltas and Gammas for Calls and Puts Stock Price, S

Call deltaC

Put deltaP

Gamma

$50

0.0000

−1.0000

0.0000

$55

0.0000

−1.0000

0.0000

$60

0.0001

−0.9999

0.0000

$65

0.0005

−0.9995

0.0002

$70

0.0030

−0.9970

0.0009

$75

0.0117

−0.9883

0.0029

$80

0.0349

−0.9651

0.0068

$85

0.0826

−0.9174

0.0126

$90

0.1620

−0.8380

0.0191

$95

0.2720

−0.7280

0.0245

$100

0.4025

−0.5975

0.0272

$105

0.5381

−0.4619

0.0265

$110

0.6636

−0.3364

0.0233

$115

0.7686

−0.2314

0.0186

$120

0.8492

−0.1508

0.0137

$125

0.9065

−0.0935

0.0094

$130

0.9446

−0.0554

0.006

$135

0.9685

−0.0315

0.0037

$140

0.9828

−0.0172

0.0021

$145

0.9909

−0.0091

0.0012

$150

0.9953

−0.0047

0.0006

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FIGURE 19.3A: Deltas for Calls and Puts Deltas 1.00 Call delta 0.50 0.00 –0.50 –1.00 Put delta 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

FIGURE 19.3B: Gamma for a Call (Same as That for the Put) Gamma

0.03 0.025 0.02 0.015 0.01 0.005 0 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

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Stock Prices

Stock Prices

THE GREEKS

439

Last, the sensitivity of the option value to changes in the interest rate is called rho: rho ≡

Change in the value of the option 𝜕c = 𝜕r Change in the value of the interest rate

(19.10e)

= TKe−rT N (d2 ) ≥ 0 As the interest rate increases, the call’s value increases. This is due to the decrease in the present value of the strike price paid at time T if the option ends up in-the-money. But a European put’s rho is negative. This happens because a rise in the interest rate lowers the present value of the exercise price that the holder might receive at expiration, and this reduces the put’s value.

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Some Road Bumps Ahead The Greeks are useful for understanding the sensitivity of option values to changes in the inputs. Sometimes the Greeks are also used for hedging option price risks caused by changes in these parameters. A warning needs to be raised at this time! We will show in Chapter 20 that delta and gamma hedging are valid procedures when the BSM assumptions are satisfied (Extension 19.2 shows what happens when the competitive markets assumption is violated and market manipulation is possible). However, even if the BSM assumptions are valid, rho hedging (which has been largely abandoned by the practitioners) and vega hedging (which should also disappear but exists today in many institutions) have serious conceptual problems. Let us drop a hint. We held volatility and interest rates constant throughout our derivation of the BSM formula, so the risk of their changing was not included in the price. This is in contrast to the stock price, whose risk was explicitly considered. Now, all of a sudden, we cannot start tinkering with something that we kept fixed. We will return to these issues in Chapter 20.

EXTENSION 19.2: Market Manipulation and Option Pricing In the derivation of the BSM model, assumption A3, the competitive and well-functioning markets assumption is crucial. Extension 19.1 already discussed how option pricing changes when stock price bubbles exist that violate the well-functioning assumption. This extension discusses the difficulties in pricing options when the competitive markets assumption is relaxed. A competitive market is one in which traders’ trades do not affect the stock price. When this assumption is relaxed and traders’ trades change the price, stock market manipulation is possible. In a market that can be manipulated, the BSM model is no longer valid. In fact, the BSM value and delta can be completely wrong. We illustrate such a dramatic failure of the BSM model with a simple example. Let us consider a European call option with strike price K and maturity T written on a stock with time t price S(t), and volatility 𝜎. Suppose that the stock price has evolved across time, as graphed in Ext. 19.2 Fig. 1, and that currently we are standing at time T−𝜀 for a small 𝜀 > 0, where the stock price is out-of-the-money, that is, S (T − 𝜀) < K

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

Using the BSM formula, it can easily be shown that the value of the call option and its delta are approximately zero, that is, c ∼ 0 and delta ∼ 0 (1) This is because the stock is trading out-of-the-money, and with so little time remaining before the option expires, it is unlikely to move much above the strike price. Now suppose that trades can change the stock price. This is the violation of the competitive market assumption. Furthermore, suppose that a manipulator purchases a large quantity of the stock’s shares purposely to cause the stock price to jump to S* > K. In light of this manipulation, the true price of the call option and its delta are easily seen to be c = S∗ − K and delta = 1

(2)

As indicated, these values are dramatically different from those predicted by the BSM model. If a risk manager had used the BSM model to hedge an option position using a delta value near zero (see Chapter 20 for an elaboration on delta hedging), he would have lost significant wealth because of the market manipulator’s actions.

EXT. 19.2 FIG. 1: Manipulated Stock Price S(t)

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S*

K Stock price

0

T–ε T

Time

this example illustrates, the BSM model no longer applies when the competitive markets assumption is relaxed. Pricing options in a world in which the stock price can be manipulated is a fruitful area for research (see Bank and Baum 2004; Jarrow 1994).

19.8

The Inputs

Most of the inputs for the BSM formula are easily obtained. The most elusive input to measure is the volatility. Because of its importance, volatility estimation is discussed in this section and also in the next chapter. This chapter concentrates on historical volatility, while Chapter 20 focuses on calibration and implied volatilities.

THE INPUTS

Observable Inputs The strike price (K) and time to maturity (T) are specified in the option contract. The stock price (S) is easy to obtain. For actively traded stocks, use the last transaction price, and for less liquid stocks, use the average of the bid and ask prices. The continuously compounded risk-free interest rate (r) is computable from Treasury security prices. Select the Treasury bill that matures closest to the option’s expiration date, take the average of the bid and ask yields, and compute the Treasury bill’s price. Last, solve for the continuously compounded annual interest rate from this zero-coupon bond price (see Chapter 2). This is the interest rate to use in the BSM formula.

Volatility: The Elusive Input

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Volatility is the hardest input to estimate. There are two standard methods for estimating the volatility: an explicit method using past stock prices, called the historical volatility, and an implicit method using option market prices, called the implied volatility. HISTORICAL VOLATILITY The volatility when estimated using statistical methods and past price data is called a historical volatility. An estimate of the volatility is the stock return’s sample standard deviation, although numerous other estimators can also be used. Historical volatility estimation, using actual stock price realizations, is based on nature’s “draws” from the actual probability distribution Equation 19.1. This implies, of course, that the estimates of both the stock’s expected return 𝜇 and volatility 𝜎 are obtained under the actual probability q. Example 19.3 illustrates how to estimate the stock’s volatility using the return’s sample standard deviation.

EXAMPLE 19.3: Computing Historical Volatility ■

Stock price data are available from newspapers or the Internet.

■

The data. Table 19.3 reports closing stock prices for Your Beloved Machines Corp. (YBM). Column 1 is labeled Week t. There are (T + 1) = 17 observations corresponding to 17 consecutive weeks, where t runs from week 0 to week 16. Column 2 records weekly closing prices for YBM. Seventeen observations are chosen to illustrate the computation. In practice, a larger number of observations are recommended. This issue is discussed after the example.

■

The price relative. The price relatives of the stock prices are reported in column 3. This is given by S(t)/S(t−1), where S(t) is week t’s closing price. For example, when t = 6, the price relative is S (6) /S (5) = 103/100 = 1.03 Notice that when computing the “difference,” the first observation is lost. We now have n =16 data points, which we denote by t = 1, 2, … ,16.

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

TABLE 19.3: Computing 𝜎 from Weekly Price Data 16

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Week (t)

■

Stock Price S(t)

Price S(t) Relative S(t − 1)

0

100

1

102

1.02

2

99

0.970588235

3

100

4

Log of Price Relative S (t) log ≡ x(t) [ S (t − 1) ]

∑

[x(t) − 𝜇]̂ 2

t =1

0.019802627

0.000344658

−0.029852963

0.000966627

1.01010101

0.010050336

7.76632E-05

101

1.01

0.009950331

7.59106E-05

5

100

0.99009901

−0.009950331

0.000125171

6

103

1.03

0.029558802

0.000802087

7

104

1.009708738

0.009661911

7.09679E-05

8

101

0.971153846

−0.029270382

0.000930741

9

101

1

0

1.53181E-06

10

100

0.99009901

−0.009950331

0.000125171

11

101

1.01

0.009950331

7.59106E-05

12

100

0.99009901

−0.009950331

0.000125171

13

102

1.02

0.019802627

0.000344658

14

102

1

0

1.53181E-06

15

99

0.970588235

−0.029852963

0.000966627

16

102

1.03030303

0.029852963

0.000818835

Sums

Σx = 0.019802627

Mean

Σx/16 = 0.001237664

0.005853263

The logarithm of the price relative. The natural logarithm of the price relative, log[S(t)/S(t−1)], is computed and recorded in the fourth column. For example, we have log [S (6) /S (5)] = log (1.03) = 0.029558802 For simplicity, write this as log[S(t)/S(t−1)] = x(t).

THE INPUTS

■

Computing the sample variance. One can proceed in several ways: - If the data are input into a spreadsheet or a computer program, the sample variance is computed in one step. For example, running Excel command VARA over the sixteen observations in cells D4 = 0.019802627 to D19 = 0.029852963 gives VARA(D4:D19) = 0.000390218. - Alternatively, one can plod through the calculations by hand, first computing the sample mean, T

𝜇̂ =

1 x (t) = 0.019802627/16 = 0.001237664 T t∑ =1

and then reporting in column 5 the deviations of the x(t)’s values from the mean, squared, and then computing the average of these squared deviations using (T − 1) = 15 in the denominator to get the sample variance: T

𝜎̂ 2 =

■

1 [x(t) − 𝜇]̂ 2 = 0.005853263/15 = 0.00390218 T − 1 t∑ =1

Computing the annualized volatility. To convert the computed sample variance to the annualized variance, because the variance is proportional to the time period (see Result 1 in the appendix to this chapter), we need only multiply by the number of weeks in a year, that is,

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𝜎̂ 2Annual = 𝜎̂ 2Week (Number of weeks) = 0.00390218 × 52 = 0.02029131 The positive square root is the annualized volatility 𝜎̂ = 0.14245. ■

Converting to annualized figures. Suppose you compute the volatility using daily data instead. How would you annualize this? Although people initially used 365 days, empirical studies suggest that a better volatility estimate is obtained if one uses the number of trading days instead (see French 1980; French and Roll 1986).3 Assuming there are 250 trading days in a year, one gets 𝜎̂ 2Annual = 𝜎̂ 2Daily (250)

3

If you compute sigma using monthly data (one stock price data per month), then use the square root of 12 as the adjustment factor to compute annualized volatility.

In computing the sample variance, a number of choices need to be made. First, how many observations should you use? Theoretically speaking, more observations will give a better estimate, everything else held constant. But everything else is almost never constant! A long time series of prices may not be stationary because the underlying structure of the company may change, and consequently, so, too, the stock’s volatility. A good rule of thumb is to get a minimum of 50 observations (where the central limit theorem from statistics kicks in, which enables the use of the normal distribution for hypothesis testing). Second, what should the time interval be between

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stock price observations? Shorter intervals imply more price observations, but at short intervals, market frictions and market microstructure issues can introduce significant noise into the estimation. Daily or weekly intervals have less of these microstructure problems. A good rule of thumb is to use daily or weekly data. IMPLIED VOLATILITY The volatility estimated using an OPM in conjunction with contemporaneous option market prices, instead of past stock prices, is called an implied volatility. This section explains how to obtain such an estimate using the BSM formula. For the purposes of this estimate, we can rewrite the BSM call price as BSM(𝜎), where 𝜎 is the volatility. The implied volatility 𝜎Imp is defined as the volatility that solves the following equation: cMarket = BSM (𝜎Imp ) (19.11)

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In words, the implied volatility is that volatility that equates the market price for an option equal to its BSM value. Thus, for the implied volatility, we use (S, K, r, T) and cMarket as inputs and then compute the volatility 𝜎 by solving this equation. Technically, since the BSM formula is an increasing function of the volatility, a solution always exists. Unfortunately, no closed form solution is available. However, using numerical approximation techniques, the option’s implied volatility 𝜎Imp can be easily determined. This estimation method is an example of calibration, in which one estimates a parameter from a model by using market prices as inputs rather than outputs. Calibration and implied volatilities are often misused, so care should be exercised when employed. Chapter 20 discusses the use and misuse of calibration and implied volatilities in detail.4

19.9

Extending the Black–Scholes–Merton Model

This section extends the BSM model in three key directions: one is to dividends, two is to foreign currency options, and three is to American options. All these extensions are important applications of the BSM model.

Adjusting for Dividends The assumption that the underlying stock pays no dividends underlies the BSM model. But most stocks pay dividends, which lowers the price of calls and raises the price of puts. Fortunately, one can adjust the basic model for dividends. We present two such approaches. The first adjusts the BSM formula for a known dollar dividend, whereas the second allows the stock to pay random dividends at a continuous rate. Here the discussion of dividend adjustments presented in the cost-of-carry model (Chapter 12) and the multiperiod binomial model (Chapter 18) are relevant. 4

Note that because implied volatility estimation uses market prices and the BSM formula, this estimation is done under the pseudo-probabilities 𝜋 and does not depend on the actual probabilities.

EXTENDING THE BLACK–SCHOLES–MERTON MODEL

When the dividend date and dollar dividend are known, replace the actual stock price with the current stock price minus the present value of all the dividends paid over the option’s life. Then continue with business as usual: employ the BSM model with this modified stock price. Thus Equation 19.4a gets modified to c = [S − PV (Dividends)] N (d1 ) − Ke−rT N (d2 )

(19.12)

where log d1 =

S − PV (Dividends) 𝜎2 + r+ T ] ( [ K 2) 𝜎√T

d2 = d1 − 𝜎√T When the dividend rate is known, again the stock price is replaced by the current stock price less the present value of the dividends paid over the option’s life, but here there is a fractional reduction (e−𝛿T ), that is, c = Se−𝛿T N (d1 ) − Ke−rT N (d2 )

(19.13a)

p = Ke−rT N (−d2 ) − Se−𝛿T N (−d1 )

(19.13b)

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where 𝛿 is the dividend yield:

d1 =

S 𝜎2 log ( ) + r − 𝛿 + T ( K 2) 𝜎√T

d2 = d1 − 𝜎√T The dollar dividend is random in this case because the random stock price is multiplied by a constant dividend rate, yielding a random dollar dividend payment. This model, originally derived by Robert Merton (1973a), is particularly useful for valuing index and currency options for the reasons we had discussed before (see Chapter 12). In the special case when 𝛿 = 0, this model reduces to the BSM model. Example 19.4 illustrates their use.

EXAMPLE 19.4: Pricing a European Option on an Index

The Data ■

Consider the data from Chapter 12: - INDY index’s current level I = 1,000. - The continuously compounded interest rate r is 6 percent per year.

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- The dividend yield 𝛿 = 20/1,000 = 0.02 or 2 percent per year because the stocks constituting INDY paid $20 in dividends last year.

The European Call Price ■

Consider a European call on INDY with maturity T = 1 year and a strike price of K = 1,100. Let the index’s return volatility 𝜎 be 0.142470 per year. The payoff to the call on the maturity date is 100[S(T) −1,100] dollars if it ends in-the-money, and is zero otherwise, where 100 is the contract multiplier. This call is valued using Equation 19.13a, as follows: - Compute d1 and d2 :

d1 =

=

𝜎2 S log ( ) + r − 𝛿 + T ( K 2) 𝜎√T (0.14247)2 1, 000 (1) + 0.06 − 0.02 + log ] ( 1, 100 ) [ 2

0.14247√1 = −0.3170 d2 = d1 − 𝜎√T = − 0.3170 − 0.14247 = −0.4595

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- Using the NORMSDIST function of Excel, we get the cumulative standard normal distribution function values N (d1 ) = N (−0.3170) = 0.3756 N (d2 ) = N (−0.4595) = 0.3230 - The zero-coupon bond price is B = e−rT = e−0.06 = 0.9418. - The present value of the dividend yield e−𝛿T = e−0.02 = 0.9802. - The European call price for the index option is therefore c = e−𝛿T SN (d1 ) − Ke−rT N (d2 ) = 0.9802 × 1, 000 × 0.3756 − 1, 100 × 0.9418 × 0.3223 = $33.6286 - Using the contract multiplier 100, the European call’s price is $3,362.86.

Foreign Currency Options Pricing foreign currency options using the BSM formula is just a clever use of the BSM model with a dividend yield. We illustrate this application with an example (see Example 19.5).

EXTENDING THE BLACK–SCHOLES–MERTON MODEL

EXAMPLE 19.5: Pricing a European Option on a Currency

The Data ■

Consider the data from Chapter 12: - Let today’s spot exchange rate SA be $2 per pound (in American terms). - Let the continuously compounded annual risk-free interest rates be r = 6 percent in the United States (domestic) and rE = 4 percent in the United Kingdom (foreign). The UK rate is interpreted as the “dividend yield” 𝛿 in this context.

The European Call Price ■

Consider a European call on pound sterling with maturity T = 1 year and a strike price of K = $2.10. Let the pound’s volatility 𝜎 be 0.10 per year. The payoff to the call on the maturity date is 100[S(T)−2.10] dollars if it ends in-the-money, and zero if otherwise, where 100 is the contract multiplier. This call is valued using Equation 19.13a: - Compute d1 and d2 : log (

(0.1)2 2 (1) + 0.06 − 0.04 + 2.1 ) [ 2 ]

= − 0.2379 0.1√1 d2 = d1 − 𝜎√T = − 0.2379 − 0.1 = − 0.3379

d1 =

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- Moreover, we get N (d1 ) = N (−0.2379) = 0.4060 N (d2 ) = N (−0.3379) = 0.3677 The zero-coupon bond price (US) is B = e−rT = e−0.06 = 0.9418. The zero-coupon bond price (foreign) BE = e −rE T = e−0.04 = 0.9608. - The European call price for the currency option is therefore c = e −rE T SN (d1 ) − Ke−rT N (d2 ) = 0.9608 × 2 × 0.4060 − 2.1 × 0.9418 × 0.3677 = $0.0529 Using the contract multiplier 100, the European currency call’s price is $5.29.

Valuing American Options The BSM formula and its dividend modifications only hold for European options. Yet, American options trade. We know from Chapter 16 that when there are dividends, American options are more valuable than European options because of an early exercise premium. Unfortunately, we can only obtain formulas for American call

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

options in some very special cases, and there are no closed-form solutions for pricing American put options. This implies that to price American options in practice, we need to turn to numerical procedures. Numerical procedures compute values using the computer instead of plugging into an analytic formula. You have already been introduced to numerical procedures in Chapter 18 when pricing American calls and puts using the binomial model. Because the multiperiod binomial model can be used to approximate a lognormally distributed stock price, the multiperiod binomial model provides an important tool for approximating American call and put values in the BSM model as well. This approach works well but can be computationally time consuming. Other numerical procedures for pricing American call and put options are available as well. Numerical approximation techniques can be divided into two categories: (1) methods that numerically solve the partial differential equation (pde) implied by the option’s price (see Chapter 20 for a discussion of the BSM pde) and (2) those that approximate the stock price evolution to compute the option’s riskneutral value (a discounted expectation). To solve pdes, numerical techniques like finite difference methods and numerical integration can be employed. To compute discounted expectations, the binomial model and Monte Carlo simulation techniques are useful. Unfortunately, both methods are beyond the scope of this textbook and are left for more advanced courses (see Glasserman [2003] for Monte Carlo methods, Wilmott [1998] for pdes, and Duffie [2001] for a discussion of both).

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EXTENSION 19.3: Exotic Options Exotic options are derivative securities that differ from the simple, plain vanilla forwards, futures, and European and American call and put options that we have encountered so far. There are many exotic options with names like “digital options,” “gap options,” “options on the maximum or minimum,” “down-and-out options,” “down-andin options,” and “average options.” The defining characteristic of any such exotic option is its payoff at maturity or at an intermediate time, conditional on certain events related to the path of the underlying stock price being satisfied. The easiest method of pricing such options is to use risk-neutral valuation and compute the discounted expected payoffs to these options under the pseudo-probabilities. We illustrate this general approach with two examples under the BSM assumptions.

Digital options A digital put option has a maturity date T and a strike price K. Its payoff at maturity is digital (T) =

1 if S (T) < K {0 if S (T) ≥ K

This payoff is similar to an ordinary European put, except in its payoff, which equals just $1 if the option ends up in-the-money.

SUMMARY

449

Using risk-neutral valuation, the time 0 value of a digital put option is digital (0) = E𝜋 {digital (T)} e−rT = e−rT N (d − 𝜎√T) where d=

S log ( ) − rT K 𝜎√T

𝜎 + √T 2

Barrier options Barrier options are similar to European options, except the payoffs depend on whether the stock price hits some barrier over the option’s life. We illustrate one such option. A down-and-out call option has a maturity date T and a strike price K. Its payoff at maturity is that of an ordinary European call option, but only if a lower barrier (b) is not hit over the option’s life. If the lower barrier is hit, the option expires worthless, regardless of its value at maturity. The payoff can be written as follows: down&out (T) = 1{𝜏 > T} max [S (T) − K, 0] where 1{𝜏>T} =

1 if min S (t) > b for all t ∈ [0, T] {0 otherwise

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Using risk-neutral valuation, the time 0 value of a down-and-out call option is down&out (0) = E𝜋 {down&out (T)} e − rT A complex analytic formula for this equation exists; see Jarrow and Turnbull (2000).

19.10 Summary 1. The year 1973 saw the publication of the Black–Scholes–Merton option pricing model. The BSM model forms the foundation of more advanced derivatives pricing models. 2. As with the binomial model, the BSM model assumes no market frictions, no credit risk, competitive and well-functioning markets, no interest rate uncertainty, no intermediate cash flows, and no arbitrage. Here, however, we introduce a different assumption for the stock price evolution. The BSM model assumes continuous trading and that the stock price returns follow a lognormal distribution.

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

3. The BSM formula for pricing a European call option is c = SN (d1 ) − Ke−rT N (d2 ) where S ≡ S(0) is today’s spot price, N( . ) is the cumulative standard normal distribution function which has mean 0 and standard deviation 1,

d1 =

𝜎2 S log ( ) + r + T ( K 2) 𝜎√T

d2 = d1 − 𝜎√T K is the strike price, r is the continuously compounded risk-free interest rate per year, T is the time to maturity measured in years, and 𝜎 is stock return’s volatility per year. 4. As in Chapter 17, it can be shown that the stock price, normalized by the price of a money market account, is a martingale under the pseudo-probabilities. This is an implication of our no arbitrage assumption. Under the lognormal distribution assumption, the market is complete, and risk-neutral valuation holds using the pseudo-probabilities and gives the call price as

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c = E𝜋 {max [0, S (T) − K]} e−rT 5. To understand the BSM formula, we compute the option sensitivities (popularly known as the Greeks) by taking partial derivatives with respect to the stock price (S) to get delta, the time to maturity (T) to get theta, the riskless interest rate (r) to get rho, and the stock return’s volatility (𝜎) to get vega. In addition, the second partial derivative with respect to the stock price is the gamma. The Greeks are useful for options hedging and portfolio risk management, a topic we address in Chapter 20. 6. Volatility is the hardest input to estimate. There are two standard methods for estimating the volatility: an explicit method using past stock prices, called the historical volatility, and an implicit method using option market prices, called the implied volatility. The implied volatility is discussed in Chapter 20. 7. Three common extensions of the BSM model are to adjust for dividends, to price foreign currency options, and to price American options. American options don’t have analytical solutions. They can be solved by numerical methods belonging to one of three types: (1) a multiperiod binomial option pricing model, (2) partial differential equations, and (3) Monte Carlo simulation techniques.

APPENDIX

19.11 Appendix Modeling the Stock Price Evolution This appendix justifies the widely used lognormal assumption for stock prices using fundamental axioms for stock price changes. We build this argument in several steps: we (1) divide the time horizon into shorter periods over which stock prices follow a continuously compounded (logarithmic) return, (2) introduce uncertainty via additional assumptions on these returns, and (3) apply the central limit theorem to show that stock prices are lognormally distributed.

Continuously Compounded (Logarithmic) Return We introduce a discrete time multiperiod structure like the one introduced in Chapter 18. We do this by chopping up the time horizon [0, T] into n shorter time periods of length T/n = ∆t each (see Figure 1). Let us assume that over any time interval [t −∆t, t], the stock price has a continuously compounded return zt that is inspired by Equation 19.1: S (t) = S (t − Δt) ezt (1) The stock price grows by continuous compounding. Setting t = ∆t, 2∆t,.. given relations like

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S (Δt) = S (0) ez1 over [0, Δt] , S (2Δt) = S (Δt) ez2 over [Δt, 2Δt] , ... S (T) = S (T − Δt) ezT over [T − Δt, T] We can use the preceding relations to write the stock price at the end of the time horizon as S (T) =

S (T) [ S (T − Δt)

S (T − Δt) S (Δt) S (0) … S (0) ] S (T − 2Δt)

= ezT ezT−1 … ez1 S (0)

(2)

= S (0) ez1 +…+zT−1 +zT Now, if we define the sum of these continuously compounded stock price returns over the short intervals as Z(T), that is, z1 + ... + zT−1 + zT ≡ Z (T)

(3a)

then we can simply rewrite (2) as Z (T) = log [S (T) /S (0)]

(3b)

Thus Z(T) corresponds to the continuously compounded (or logarithmic) return over the time horizon [0, T]. Notice that (3a) is a linear relationship that has a number of attractive features and makes the model amenable to mathematical analysis.

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FIGURE 1: Modeling the Stock Price Evolution Stock price

S

0

∆t

T – ∆t T

2∆t

Time

This graph divides the time horizon [0, T] into n intervals of length ∆t each.

Introducing Uncertainty Next, we introduce uncertainty in our model. We do this by adding two assumptions, which are often-used statistical properties for a collection of random variables.

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A1. The returns {zt } ≡ {z1 , z2 , … , zT } are independent. Early empirical studies suggested that stock price returns over successive intervals are independent of each other. A2. The returns {zt } are identically distributed. Early empirical studies also suggested that stock price returns over successive intervals tend to follow the same distribution. Notice that we have not specified any probability distribution for the continuously compounded stock price returns. These two assumptions yield the property that is often referred to by the abbreviation iid (for independent and identically distributed). Any process satisfying these assumptions is called a random walk. Stock prices following a random walk are linked with the efficient markets hypothesis introduced in Chapter 6 because random walks occur when information comes in randomly and gets immediately reflected in stock prices. Next, we introduce two more assumptions to ensure that stock prices behave as the time interval shrinks and approaches zero. We do this by assuming that the expected return and the variance of the return are proportional to the length of the time interval. A3. The expected continuously compounded return can be written as E (zt ) = 𝜇Δt

(4)

where 𝜇 is a constant that corresponds to the expected continuously compounded return per year.

APPENDIX

A4. The variance of the continuously compounded return can be written as var (zt ) = 𝜎2 Δt

(5)

where 𝜎2 is a constant that corresponds to the variance of the continuously compounded return per year. Assumptions A1−A4 allows us to write E [Z (T)] = E (z1 + … + zT−1 + zT ) = 𝜇 (Δt + Δt + ... + Δt) [because, by A3, the returns have identical mean 𝜇] = 𝜇T

(6a)

and var [Z (T)] = var (z1 + ... + zT–1 + zT ) = var (z1 ) + var (z2 ) + ... + var (zT )

[by assumption A1]

n

=

∑

𝜎2 Δ

[by assumption A4]

(6b)

j=1

= 𝜎2 T

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We used expression (6b) earlier in the chapter when estimating the stock’s historical volatility.

The Central Limit Theorem Next, we apply the central limit theorem (CLT). It states that the average of iid random variables (provided it has a well-defined mean and variance), when “standardized” (by subtracting from the average the common mean and dividing the expression by the standard deviation of the average), starts behaving like a standard normal distribution (see Mood et al. 1974; Cox and Miller 1990). An application of this theorem yields the following result (see Result 1).

RESULT 1 Lognormal Distribution of Stock Price Returns Stock prices are lognormally distributed with mean 𝜇T and variance 𝜎2 T over the time horizon [0, T].

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

Hence we can write the continuously compounded return as Z (T) = log

S (T) 𝜎2 = 𝜇− T + 𝜎√Tz ( S (0) ) ( 2)

(7)

where z is a standard normal random variable with mean 0 and variance 1. Because z is normally distributed, Z(T) is normally distributed, and hence S(T) is said to have a lognormal distribution. This result is reported as Equation 19.2 in the text, and it is used in Example 19.3.

First Derivation of the Black–Scholes–Merton Formula (as a Limit of the Binomial Model) This appendix provides a formal derivation of the BSM formula based on the intuition obtained from Figure 19.1, the second graph of the multiperiod binomial model. In this figure, you can see that as the time interval becomes smaller, the multiperiod model appears more and more realistic. In fact, it can be shown that it approaches the lognormal distribution! Not surprisingly, this gives us a way to derive the BSM formula.

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MULTIPERIOD BINOMIAL PARAMETERS FOR THE LOGNORMAL APPROXIMATION ■

To formalize this intuition, we start with assumption A7 of Chapter 17 as applied to the multiperiod binomial model. We divide our time interval [0, T] into subintervals of length ∆t, yielding T/∆t = n intervals.

■

Next, we use the Jarrow–Rudd specification of the parameters rΔt −

U=e

𝜎2 𝜎2 Δt + 𝜎√Δt Δt − 𝜎√Δt rΔt − 2 2 and D = e

(8a)

given in Extension 18.1. Because the option value does not depend on the actual probability of moving up, we leave the actual probability unspecified, and work under the pseudo-probabilities. We choose the pseudo-probability 𝜋 = 1/2. This choice guarantees that the binomial process for the log return on the stock over [0, T] has an expectation and volatility equal to rT − (𝜎2 /2)T and 𝜎2 T, respectively. Using expression (8a), the stock price process is S (T) = S (0) U j D(n−j) ■

(8b)

Taking logarithms of expression (8b) yields log S (T) = log S (0) + j log U + (n − j) log D

(9a)

APPENDIX

Next, substitution of U and D into this expression gives 𝜎2 log S (T) = log S (0) + j rΔt − Δt + 𝜎√Δt ( ) 2 2 𝜎 + (n − j) rΔt − Δt − 𝜎√Δt ( ) 2

(9b)

Substituting ∆t = T/n and simplifying this is transformed to log S (T) = log S (0) + 2j

𝜎√T ( √n )

𝜎2 + rT − T − √n𝜎√T ( 2 )

(9c)

Last, rearranging terms gives log

S (T) 𝜎2 − rT − T 2 ) s (0) ( 𝜎√T

n (j − 2 ) = √n 2

(9d)

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TAKING LIMITS ■ The famous de Moivre–Laplace theorem states that a binomial random variable approaches a normally distributed random variable as the number of realizations (time intervals) approaches infinity, that is, ⎛ ⎞ n ⎜j − ⎟ 2 ≤ x⎟ → N (x) as n → ∞ Prob ⎜⎜ ⎟ ⎜ √n ⎟ ⎝ ⎠ 2

(10a)

Hence, by substitution of expression (9d), we see that the logarithm of the assumed stock price process approaches a normally distributed random variable, that is, S (T) 𝜎2 ⎛ ⎞ ⎜ log S (0) − (rT − 2 T) ⎟ Prob ⎜⎜ ≤ x⎟⎟ → N (x) as n → ∞ ⎜ ⎟ 𝜎√T ⎝ ⎠

(10b)

In simpler notation, this implies that as n → ∞, (

S (T) = S (0) e

r−

𝜎2

T+𝜎√Tz 2 )

(11)

where z is a standard normal random variable with mean zero and unit variance under the pseudo-probability 𝜋(z). This is our lognormally distributed stock price!

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

MARTINGALE PRICING ■

The normal distribution has the property that 𝜎2 𝜋

𝜎√Tz

E [e

]=e2

T

(12a)

Taking the expectation of expression (11) yields

E𝜋 [S (T)] =

𝜎2 T r− ( 2 ) E𝜋 S (0) e

𝜎√Tz

[e

]

(12b)

The use of expression (12a) in (12b) yields, after rearranging terms, the martingale pricing relation expression (19.6) given earlier in the text, that is, S (0) = E𝜋 [S (T)] e−rT

(13)

where E𝜋 ( . ) is expectation using the pseudo-probabilities 𝜋(z) associated with the lognormal distribution; that is, the current stock price S(0) is the expected discounted value of the stock price at time T. This is analogous to expression (17.5).

RISK-NEUTRAL VALUATION

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■

Next, using the multiperiod binomial call option formula from Chapter 18, Equation 18.11, with the Jarrow–Rudd parameters, we have n

cn = e−rT

n j 𝜋 (1 − 𝜋)n−j × [max (0, Uj Dn−j S (0) − K)] ∑ {[( j ) } ]

(14)

j=0

■

Using expression (8b) again, this can also be written as (see [18.12]) cn = E𝜋 {max [0, S (T) − K]} e−rT

(15)

This is called risk-neutral valuation.

TAKING LIMITS AGAIN ■

Taking the limit as n → ∞, one gets cn → E 𝜋 {max [0, S (T) − K]} e−rT

(16)

where 𝜋(z) are the pseudo-probabilities associated with the random variable z as in expression (11). This follows from interchanging the limit and expectation operations (see Billingsley 1995, p. 291). The next section shows that this yields the BSM formula.

APPENDIX

Second Derivation of the Black–Scholes–Merton Formula (Using Risk-Neutral Valuation) Using expression (16) above, we have c = E𝜋 {max [0, S (T) − K ]} e−rT

(17)

EVALUATING THE EXPECTATION Taking conditional expectations, one can write the call value as the sum of two terms:5 c = E𝜋 {max [S (T) − K, 0] |S (T) ≥ K} Prob𝜋 {S (T) ≥ K} e−rT +E𝜋 {max [S (T) − K, 0] |S (T) < K} Prob𝜋 {S (T) < K} e−rT

(18)

Recognizing that if S(T) ≥ K, max[0, S(T)−K] = S(T) − K, and if S(T) < K, max[0, S(T) − K] = 0, we can rewrite the last expression as c = E𝜋 {S (T) |S (T) ≥ K} e−rT Prob𝜋 {S (T) ≥ K} − KProb𝜋 {S (T) ≥ K} e−rT (19) One can show using the properties of normal distributions (see Jarrow and Rudd 1982) that E𝜋 {S (T) |S (T) ≥ K} Prob𝜋 {S (T) ≥ K} e−rT = S (0) N (d1 ) KProb𝜋 {S (T) ≥ K} e−rT = KN (d2 ) e−rT

(20)

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Combined, these give the BSM formula. This completes the derivation.

The Probabilities and the Risk Premium In our derivation of the BSM formula, we showed that one could write the stock price today as its discounted value using the pseudo-probabilities, that is, S (0) = E𝜋 [S (T)] e−rT

(21)

where E𝜋 represents expectation using 𝜋(z). Recall that the actual and pseudo-probabilities are related by the expression 𝜋 (z) = 𝜙 (z) q (z) where 𝜙(z) = e−(𝜃

2

/2)+𝜃z

(22)

> 0 and 𝜃 = − [(𝜇 − r) /𝜎] √T. Using 𝜙(z), we can rewrite

S (0) = E [𝜙 (z) S (T)] e−rT ∞

= ∫ 𝜙 (z) S (T) q (z) dz e−rT [ −∞ ]

5

(23)

This follows from the fact that E (X) = ∑Y E (X|Y) Prob (Y), where X and Y are random variables.

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CHAPTER 19: THE BLACK–SCHOLES–MERTON MODEL

This expression transforms the expectation taken in terms of the pseudo-probabilities to an expectation taken in terms of the actual probabilities. There are two ways to compute a present value. One is as given in expression (21). Here the present value is computed by taking an expectation using the pseudoprobabilities and discounting at the risk-free rate. The adjustment for risk occurs in the use of the pseudo-probabilities when taking the expectation. The second and traditional way of computing a present value is to take the expectation using the actual probabilities and adjust the discount rate for risk. Expression (23) can be manipulated to generate this second method of computing a present value. Indeed, as proved immediately, this expression can be equivalently written as ∞

S(T) −(rT+cov[log( ,− log 𝜙(z)] S(0) )

S (0) = ∫ S (T) q (z) dz e ] [ −∞ −(rT+cov[log(

= E [s (T)] e

(24)

S(T) ,− log 𝜙(z)]) S(0) )

where the discount rate is adjusted instead by the covariance term cov(log [S(T)/S(0)], −log [𝜙(z)]), a risk premium. We see that the risk-adjusted discount rate depends on 𝜙(z), which provides an alternative confirmation that 𝜙(z) is an adjustment for risk.

Proof of (24)

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This proof just uses simple calculus and knowledge of normal distributions. We start with ∞ S (0) = ∫ 𝜙 (z) S (T) q (z) dz e−rT [ −∞ ]

(25)

Plugging in 𝜙(z) and S(T) gives ∞

−

S (0) = S (0) ∫ e

𝜃2 2

+ 𝜃z

z2 𝜇T− T + 𝜎√Tz e 2 2 𝜎2

−

e

√2𝜋

−∞ ∞ 𝜇T−rT + 𝜎√T𝜃

= S (0) e

−(𝜎√T + 𝜃−z) 2

∫ e

1 2

dz e−rT dz

√2𝜋

−∞

which we get by completing the square. However, ∞

∫ e

−(𝜎√T + 𝜃−z) 2

−∞

1 2

dz √2𝜋

=1

since this is normal with mean (𝜎√T + 𝜃) and variance 1. Hence S (0) = S (0) e𝜇T−rT+𝜎√T𝜃 , which implies 𝜇T − rT + 𝜎√T𝜃 = 0 or 𝜃 = − [(𝜇 − r) /𝜎] √T

QUESTIONS AND PROBLEMS

Last, S (T) cov log , − log [𝜙 (z)] ( [ S (0) ] ) 𝜎2 𝜃2 = cov 𝜇T − T + 𝜎√Tz, − 𝜃z [ ] 2 2 = −𝜎√T𝜃cov (z, z) = −𝜎√T𝜃 because cov(z, z) = var(z) = 1. Thus 𝜇T = rT + cov

S (T) log , − log [𝜙 (z)] ( [ S (0) ] )

Finally, note that under q(z), from expression (13) above S (0) = E [S (T)] e−𝜇T and substitution completes the proof.

19.12 Cases Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

MW Petroleum Corp. (A and B) (Harvard Business School Cases 295029 and

294050-PDF-ENG). The cases focus on evaluation and execution of a creative financing structure that allows reallocation of oil price risk in the context of an acquisition. MoGen Inc. (Darden School of Business Case UV1054-PDF-ENG, Harvard Busi-

ness Publishing). The case shows how to price a convertible bond offering as a straight bond plus the conversion option. Sara’s Options (Harvard Business School Case 201005-PDF-ENG). The case stud-

ies the valuation of stock options in pay packages offered to a graduating MBA student.

19.13 Questions and Problems 19.1. Explain why pricing and hedging go hand in hand in the Black–Scholes–

Merton option pricing framework. 19.2. What does the Black–Scholes–Merton model assume about the evolution of

the stock price, about dividends paid on the stock, and about interest rates? 19.3. Do you think the Black–Scholes–Merton model assumptions of no dividends

paid on the stock and constant interest rates are reasonable assumptions? If not, why do you think they are imposed?

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19.4. In the assumption of the Black–Scholes–Merton model, a. what is the stock’s return volatility? b. what is assumed about the stock’s return volatility? Is this assumption

reasonable or not? If not, why do you think it is used? The next two questions use the following information for pricing European options. Special Motors Corporation’s stock price S is $59, the strike price K is $60, the maturity T is forty-four days, the implied volatility 𝜎 is 30 percent per year, and the risk-free interest rate r is 3.3 percent per year. 19.5. Using the Black–Scholes–Merton formula, find the price of a call option. 19.6. Using the Black–Scholes–Merton formula, determine the price of a put

option. 19.7. Verify whether the options prices obtained in the previous two questions

satisfy put–call parity. 19.8. Consider two investors who agree on the stock’s price and volatility but who

do not agree on the stock’s expected return. One believes that the stock price will earn 15 percent over the next year, while the second believes that it will have a negative 5 percent return. Will they agree or disagree on the value of a one-year call option on the stock when using the Black–Scholes–Merton model? Explain your answer. 19.9. Explain how to compute the Black–Scholes–Merton model using the pseudo-

probabilities. What is this principle called? 19.10. What is the difference between the actual probabilities and the pseudoCopyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

probabilities, and on what does it depend? Explain your answer. 19.11. What are the Greeks in the Black–Scholes–Merton formula? Which two

Greeks correspond to changes in the stock price? 19.12. Comment on the following questions, carefully explaining your answers. a. As the stock price input increases, everything else constant, does a call

option’s price increase or decrease? b. As the volatility input increases, everything else constant, does a call

option’s price increase or decrease? 19.13. Given the data for Special Motors Corp. given earlier (i.e., S is $59, K is $60,

T is forty-four days, 𝜎 is 30 percent per year, and r is 3.3 percent per year), compute the call option’s delta and gamma. 19.14. a. Which inputs to the Black–Scholes–Merton model are observable and

which need to be estimated? b. Describe some of the difficulties in estimating the unobservable input. 19.15. What is the implied volatility when using the Black–Scholes–Merton model?

Does this estimate depend on the stock’s time series of past stock prices? Explain your answer.

QUESTIONS AND PROBLEMS

19.16. FunToy’s stock has the following weekly closing prices: $40, 41, 43, 42, 42,

46, 43, 44, 47. Assuming fifty-two trading weeks in a year, compute the annualized, sample standard deviation (volatility) 𝜎. 19.17. Fallstock Inc.’s stock price S is $45. European options on Fallstock have a

strike price K = $50, a maturity T = 180 days, the risk-free interest rate r = 5 percent per year, and the let the volatility 𝜎 = 0.25 per year. The company is going to pay a dividend of 50 cents after 125 days. a. Compute the price of a call option using the Black–Scholes–Merton

model. b. Compute the price of a put option. 19.18. SINDY index is currently at I = 11,057. European options on SINDY have

a strike price K = 11,000, a maturity T = 45 days, a dividend yield 𝛿 = 1.5 percent per year, the risk-free interest rate r = 5 percent per year, and the estimated implied volatility 𝜎 = 16 percent per year. a. Compute the price of a call option using the Merton formula. b. Compute the price of a put option using the appropriate version of put–

call parity (see Chapter 16).

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19.19. A European call on euro matures after T = 6 months. The call pays on the

maturity 100[S(T) − 30] dollars if it ends in-the-money, and zero is otherwise, where 100 is the contract multiplier and $1.30 is the strike price K. Euro’s volatility 𝜎 is 12 percent per year. Today’s spot exchange rate SA is $1.4 per euro (in American terms). The continuously compounded annual risk-free interest rates are r = 5 percent in the United States (domestic) and rE = 4.5 percent in the Eurozone. Compute the price of the call option using the Merton formula. 19.20. How would you value an American call option in the Black–Scholes– Merton

model?

461

20

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Using the Black–Scholes–Merton Model 20.1 Introduction 20.2 Hedging the Greeks

20.6 The Black–Scholes–Merton Model: A Postscript

Delta Hedging

20.7 Summary

Gamma Hedging

20.8 Appendix

20.3 Hedging a portfolio of options 20.4 Vega Hedging EXTENSION 20.1 Stochastic Volatility Option Models

20.5 Calibration Theoretical and Econometric Models Using Calibration Implied Volatilities EXTENSION 20.2 VIX: The Fear Index

The Mathematics of Delta, Gamma, and Vega Hedging A Third Derivation of the Black–Scholes–Merton Formula

20.9 Cases 20.10 Questions and Problems Overview Data (from the Wall Street Journal, March 3, 2000) The Project

HEDGING THE GREEKS

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20.1

Introduction

More than a century ago, financiers who sold options for strategic and manipulative purposes dominated the New York City options market. In contrast, London dealers acted more like insurance agents employing a “model” to estimate the option’s premium as the stock’s average price less the strike price plus various costs and profits. These sellers would also often “delta hedge” their positions. New Yorker Samuel A. Nelson (1904, p. 14) remarked that the London options market, “where the business is more of a science” prospered more than its Wall Street counterpart. Isn’t it interesting that a model helped organize a trader’s thinking, placed trading on a more scientific footing, and helped the market grow more than a century before our time! Today, models are actively used in financial markets. In fact, model misuse and abuse significantly contributed to the 2007 financial crisis. Many financial institutions misused models and, consequently, underestimated the risk of their positions. These errors led to insufficient equity capital in the economy, resulting in the failures of numerous financial institutions. If the financial institutions had better understood the model subtleties, perhaps this credit crisis could have been avoided. Understanding the content of this chapter will set you on the path to avoiding such errors. This chapter explains how to use the Black–Scholes–Merton (BSM) model to facilitate risk management. We show that both delta and gamma hedging are consistent with options pricing theory. We also explain why two other once popular techniques, the now-discarded rho hedging and the still-practiced vega hedging, are not. Next we revisit the knotty problem of volatility estimation via the computation of implied volatilities. This is an example of calibration, a popular technique that also has significant scope for abuse. We end this chapter with a discussion of the BSM model’s extensions and generalizations.

20.2

Hedging the Greeks

Hedging the Greeks is a risk management technique used to eliminate market risks from a stock or options portfolio. Borrowing terms from the title of the classic western movie The Good, the Bad and the Ugly, we label delta and gamma hedging as “the good” and faulty practices like vega and rho hedging as “the bad and the ugly.” Both delta and gamma hedging are “the good” because they are consistent with the theory (and assumptions) underlying the BSM model. In contrast, vega and rho hedging are “the bad and the ugly” because they are inconsistent with the theory. Hence, as argued later, they do not and, in fact, cannot fulfill their desired objectives.

Delta Hedging The delta (or hedge ratio) is the number of shares of the stock to trade for each option held to remove all price risk from an option position. For this reason, in earlier chapters, we characterized the hedge ratio as the holy grail of options pricing.

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CHAPTER 20: USING THE BLACK–SCHOLES–MERTON MODEL

To construct a delta-hedged position, we consider a portfolio V (t) at time t consisting of a long call with price c(t) and n shares of the underlying stock worth S(t) each. Suppressing the time symbol t for simplicity, the initial value of the portfolio is V = c + nS

(20.1a)

The change in this portfolio’s value over a small time interval [t, t + ∆t] can be written as ΔV = Δc + nΔS

(20.1b)

where ∆V ≡ V (t + ∆t) – V (t), ∆c ≡ c(t + ∆t) – c(t), and ∆S ≡ S(t + ∆t) – S(t). The goal of delta hedging is to choose n such that the variance of the change in the portfolio’s value is equal to zero, that is, var(∆V ) = 0, where var( . ) denotes variance. Then, the portfolio will be riskless. Recognizing that the call’s price is a function of time and the underlying stock price, using a Taylor series expansion from first-year college calculus, one can solve for this n (see the appendix to this chapter). The solution is the option’s delta. We state this observation as a result.

RESULT 20.1

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Delta Hedging in the Black–Scholes–Merton Model To remove price risk from a long call position, short deltaC ≡ where d1 =

𝜕c = N (d1 ) shares of stock 𝜕S 𝜎2 S log ( ) + r + T ( K 2) σ√T

(20.2a)

(20.2b)

N( . ) is the cumulative standard normal distribution function (which has mean 0 and standard deviation 1), S is the stock price, K is the strike price, r is the continuously compounded risk-free interest rate, T is the time to maturity, and 𝜎 is the stock return’s volatility (see Result 19.1). To delta hedge a long put position, buy (–deltaP ) ≡ N(–d1 ) stocks.

We illustrate the effectiveness of this result with a numerical example.

HEDGING THE GREEKS

EXAMPLE 20.1: Delta Hedging ■

Suppose we buy a European call option on Your Beloved Machine’s (YBM) stock at time t by paying a price c. We assume that the stock price S is $100, the strike price Kis $110, the time to maturity T is 1 year, the volatility 𝜎 of the stock’s return is 0.142470 per year, and the continuously compounded risk-free interest rate r is 5 percent per year.

■

To hedge one long call position, short deltaC shares of YBM. This gives a delta-neutral portfolio V = c – (deltaC )S. We compute the values of ∆V = ∆c – (deltaC )∆S for various stock prices in Table 20.1 to see how the delta hedge works. For simplicity, we assume that the stock price can magically change without changing any of the other inputs to the model!1

TABLE 20.1: Delta Hedging Errors for Calls and Puts

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S

d1

DeltaC = N(d1 )

DeltaP = −N(–d1 )

Call

Put

∆V (Call)

∆V (Put)

$90

–0.9863

0.1620

0.8380

$1.0294

$15.6646

$1.2709

$1.2709

95

–0.6068

0.2720

0.7280

2.1030

11.7382

0.3319

0.3319

99

–0.3173

0.3755

0.6245

3.3948

9.0300

0.0135

0.0136

100

–0.2468

0.4025

0.5975

3.7838

8.4190

0

0

101

–0.1770

0.4298

0.5702

4.1999

7.8351

0.0136

0.0136

105

0.0957

0.5381

0.4619

6.1367

5.7719

0.3403

0.3403

110

0.4222

0.6636

0.3364

9.1477

3.7829

1.3386

1.3386

- Row 4 reports S = $100 in column 1, d1 = −0.2468 in column 2, N(d1 ) = deltaC = 0.4025 in column 3, and the call price $3.7838 in column 5 (see Example 19.2). - Column 5 reports call values corresponding to the stock prices recorded in column 1. - Column 7 reports changes in the hedged portfolio’s value for different stock prices: ΔV = Δc − (deltaC ) ΔS = [c (t + Δt) − c (t)] − (deltaC ) [S (t + Δt) − S (t)] = (New call price − 3.7838) − 0.4025 (New stock price − 100) For example, for S = $95, we get ∆V = (2.1030 – 3.7838) – 0.4025(95 – 100) = $0.3319.

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■

As seen in Table 20.2, delta hedging works well for small movements in the stock price. Indeed, ∆V is smaller when the new stock price is close to the original stock price (e. g., when S = 99, ∆V = 0.0135), and it has larger values when it deviates further (e. g., when S = 110, ∆V = 1.3386).

■

Following the same approach, one can make a long put trade delta-neutral by buying n = –deltaP = –[–N(–d1 )] = 0.5975 shares. This is reported in row 4 of Table 20.2. - We get a delta-neutral portfolio V = p + (deltaP )S at time t. - Using values from column 6 of Table 20.2, the change in value of this portfolio over [t, t + ∆t] is ΔV = Δp + (deltaP ) ΔS = (New put price − 8.4190) + 0.5975 (New stock price − 100)

For S = $95, we get ∆V = (11.7382 – 8.4190) + 0.5975(95 – 100) = $0.3319. As before, the hedge works well for small fluctuations in the stock price. 1

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This gives a rough-and-ready estimation of hedging performance. For better approximations, you can track ∆V over a longer period (such as a day) and make allowance for the interest cost V r∆t.

Aren’t these delta-neutral trades akin to the simple 1:2 hedges (which combine a long stock with two short calls or two long puts) that we discussed in chapter 15? In terms of actual implementation of the position, delta hedges are minor variants of such trades. However, the difference lies in the trader’s objective as well as in her performance. Delta hedging, in theory, creates portfolios that are riskless for small fluctuations in the underlying stock price, as the previous example illustrates, whereas the 1:2 hedges are not riskless.

Gamma Hedging “In theory, there is no difference between theory and practice. But, in practice, there is,” mused baseball player Yogi Berra. This quote applies well to delta hedging. If the world behaved as per the BSM assumptions, then hedging using delta is sufficient to make an option position riskless. One can continuously rebalance the hedge to perfectly replicate the option’s payoff. The theory assumes, as noted, that the time between rebalancing, ∆t, is infinitesimal, and therefore the corresponding changes in the stock price are small. But continuous trading is impossible. For weekly or even daily hedging, changes in the stock price are not small, and the hedging error can be significant. In addition, there are the familiar transaction costs that accompany any trade, which also makes continuous trading prohibitively expensive. To address this problem, we need to hedge both small and large changes in the stock price over the hedging interval. We do this by also hedging the risk from changes in the “stock price squared.” The second hedge uses the option’s gamma, obtained by taking the second derivative of the BSM call price with respect to the stock price. The objective of the second hedge is to anticipate changes in the delta itself (the delta

HEDGING THE GREEKS

changes with changes in the underlying stock price) and to compensate for these changes. To hedge these two risks, we need two securities so that we can establish a gammahedge in addition to a delta hedge. We can hedge a short call by trading the underlying stock and another option on the same stock. Consider forming a portfolio V (t) consisting of one long call with price c1 (t), n1 shares of the stock with price S(t), and n2 units of the second option with price c2 (t). The value of the portfolio is V = c1 + n1 S + n2 c2

(20.3a)

The change to this portfolio’s value over the time interval [t, t + ∆t] is ΔV = Δc1 + n1 ΔS + n2 Δc2

(20.3b)

The objective of delta and gamma hedging is to choose n1 and n2 such that the variance of the position is zero, that is, var(∆V ) = 0. Using a Taylor series expansion, the appendix to this chapter shows that the solution is as follows.

RESULT 20.2

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Delta and Gamma Hedging a Long Call Position To remove delta and gamma price risk from a long call with price c1 , hold n1 shares of stock with price S and n2 shares of another option on the same stock with price c2 , where n1 = (gammaC1 /gammaC2 ) deltaC2 − deltaC1

(20.4a)

and n2 = −gammaC /gammaC

(20.4b)

1

2

The Greeks are defined by the general formulas

deltaC ≡

𝜕c = N (d1 ) where d1 = 𝜕S and gammaC ≡

S 𝜎2 log ( ) + r + T ( K 2) σ√T

(20.4c)

𝜕2 c 1 = N′ (d1 ) 2 𝜕S Sσ√T

where N ′( . ) is the cumulative standard normal distribution function and N ′ ( . ) is the standard normal density function.

The next example illustrates the effectiveness of both gamma and delta hedging.

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EXAMPLE 20.2: A Gamma- and Delta-Neutral Portfolio ■

Suppose we buy a European call on YBM at time t for c1 dollars. From the previous example, S = $100, K = $110, T = 1 year, 𝜎 = 0.142470 per year, and r = 0.05 per year.

■

Let us delta and gamma hedge this position by trading n1 shares of YBM and n2 shares of a second call also on YBM. Let the second European call’s price be c2 , with a strike price of $105 and a maturity of nine months.

■

The change in this portfolio’s value over the time interval ∆t is given by ∆V = ∆c1 + n1 ∆S + n2 ∆c2 .We show the performance of this hedge in Table 20.2.

■

In row 4, columns 2–5 use S = $100, K = $110, T = 1 year, 𝜎 = 0.142470, and r = 0.05 to compute the relevant values for call 1: (d1 )C1 = – 0.2468 in column 2, deltaC1 = N ′ (d1 )C1 = 0.4025 in column 3, Call1 = $3.7838 in column 4, and gammaC1 = N ′ (d1 )/(S𝜎√T) = 0.0272 in column 5.

■

In row 4, columns 6–9 do the same for call 2 with the strike price 105 and time to maturity 0.75 years: (d1 )C2 = –0.0298 in column 6, deltaC2 = N(d1 )C2 = 0.4881 in column 7, Call2 = $4.4000 in column 8, and gammaC2 = N ′ (d1 )/(S𝜎√T) = 0.0323 in column 9.

TABLE 20.2: Delta and Gamma Hedging Errors for Calls

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S

■

(d1 )C1

DeltaC1 = N(d1 )

Call1

GammaC1

(d1 )C2

DeltaC2 = N(d1 )

Call2

GammaC2

∆V

$90 –0.9863

0.1620

1.0294

0.0191

–0.8837

0.1884

$1.0857

0.0243

–$0.0092

95 –0.6068

0.2720

2.1030

0.0245

–0.4455

0.3280

2.3630

0.0308

–0.0048

99 –0.3173

0.3755

3.3948

0.0269

–0.1113

0.4557

3.9281

0.0325

–0.0001

100 –0.2468

0.4025

3.7838

0.0272

–0.0298

0.4881

4.4000

0.0323

101 –0.1770

0.4298

4.1999

0.0273

0.0508

0.5203

4.9043

0.0320

0.0000

105 0.0957

0.5381

6.1367

0.0265

0.3656

0.6427

7.2344

0.0288

0.0117

110 0.4222

0.6636

9.1477

0.0233

0.7427

0.7712

10.7826

0.0223

0.1154

Using the formulas in Result 20.2, we compute n1 = (gammaC /gammaC ) deltaC2 − deltaC1 = 0.007684 1 2 n2 = −gammaC /gammaC = −0.840422 1

2

0

HEDGING A PORTFOLIO OF OPTIONS

■

Column 10 reports changes in the delta- and gamma-hedged portfolio’s value for different stock prices over a short time interval: ΔV = Δc1 + n1 ΔS + n2 Δc2 = [c1 (t + Δt) − c1 (t)] + n1 [S (t + Δt) − S (t)] + n2 [c2 (t + Δt) − c2 (t)] = (New call price1 − 3.7838) + 0.007684 (New stock price − 100) −0.840422 (New call price2 − 4.4000) For S = $95, we get ∆V = (2.1030 – 3.7838) + 0.007684(95 – 100) – 0.840422(2.3630 – 4.4000) = – $0.0048.

■

This is less than the hedging error 0.3310 that we found for a corresponding stock price change in Example 20.1 where we only delta hedged.

■

A comparison of the last columns in Tables 20.2 and 20.3 indicates the superiority of delta and gamma hedging over delta hedging alone.

20.3

Hedging a portfolio of options

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The magic of finance is that what works for a single security can usually be extended to a portfolio of securities. Using this magic, we extend the idea of delta and gamma hedging to a portfolio of options on one or more underlying stocks. Let’s begin with a simple example.

EXAMPLE 20.3: Computing the Delta of a Portfolio Consisting of Two Options on a Single Stock Suppose that you create a diagonal spread (see Chapter 15) by buying a hundred one-year YBM 110 calls and writing the same number of nine-month YBM 105 calls. When the stock price is $100, we compute their deltas, as reported in the previous example, where deltaC1 = 0.4025 and deltaC2 = 0.4881, respectively. The portfolio delta is the number of options times the deltas, then summed, that is, deltaPort = 100 × 0.4025 − 100 × 0.4881 = −8.5575 The portfolio’s delta implies that the portfolio’s value will decline by approximately $8.56 for a dollar increase in stock price. We can generalize this example to develop rules for portfolio risk management.

■

Computing the delta ( and gamma) of an option portfolio on a single stock. Suppose that you are holding ni options (ni is positive for long holdings, negative for short) where i = 1, . . . , N denotes European calls or puts on YBM with different strikes and expiration dates. The delta of this portfolio is just the number of shares

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of each option times the deltas of each of the options summed across all the options: N

deltaPort =

∑

ni × deltai

(20.5)

i =1

You can similarly compute the gamma of an option portfolio on a single stock as the weighted sum of the individual gammas, where the weights are the share holdings. ■

Delta (and gamma) hedging an option portfolio on multiple stocks. In this case, you need to delta hedge the option portfolio with respect to each of the different underlying stocks. - First, partition the portfolio based on the underlying stocks into subportfolios. - Next, compute the delta of each subportfolio by using expression (20.5). - Finally, delta hedge each subportfolio with respect to its underlying stock. - Analogously, you can extend this line of argument to gamma hedge a portfolio of options.

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20.4

Vega Hedging

When constructing the BSM model, by assumption, there is only one risk: the risk of stock price changes. Consequently, the OPM reflects the pricing of this risk. No other risks are present in the model’s formulation. Delta and gamma hedging using the BSM model are consistent with the theory underlying the BSM model because these quantities reflect the sensitivity of the option’s value to stock price risk. The inability to trade continuously is what makes gamma hedging useful. All the other inputs to the BSM model, the interest rate (r) and the volatility (𝜎), are assumed to be constant. The risks of changing interest rates and volatility are therefore not reflected in the BSM formula. It is nonsensical to hedge changes in these constant parameters using the Greeks derived from the BSM formula. A simple example illustrates why this statement is true. After studying the example, we will apply this insight to vega hedging.

EXAMPLE 20.4: Errors in Hedging a Constant Parameter ■

Consider a European call with strike price K and expiration date T that trades on Your Beloved Machines (YBM). We will trade YBM stocks to delta hedge a long position in this call option.

A Stock Growing at a Constant Rate ■

Let us perform a thought experiment and introduce an unrealistic assumption. Let’s assume that today’s stock price S(t) is nonrandom and grows at the risk-free rate r. We agree that this assumption is silly, but is it any more silly than assuming that the stock’s return has a constant volatility?

VEGA HEDGING

■

Continuing, given this assumption, at time t, we know whether the option is going to end in-themoney. Discounting the call’s known time T payoffs at the risk-free rate gives the arbitrage-free call price: c = max [S − Ke−r(T−t) , 0] (20.6)

■

There is no need to hedge our option position because there is no risk.

Delta Hedging Using Expression (20.6) ■

We notice, however, that when applied to market data, the stock price is stochastic and our nonhedge doesn’t work. The hedging error is the entire change in the option’s price ∆c.

■

As an ad hoc fix, we attempt to hedge the Greek. In this case, the stock price is the relevant parameter, so we want to delta hedge using expression (20.6).

■

As justification, we employ the Taylor series expansion of the call formula (20.6) around the stock price over [t, t + ∆t], as discussed previously. We delta hedge by choosing n shares of the stock to short for each long call, where 𝜕c (20.7) n= 𝜕S For this model, expression (20.6), we can compute the option’s delta: n=

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■

1 if S − Ke−r(T−t) > 0 𝜕c = 𝜕S {0 otherwise

(20.8)

Here the delta hedge is to hold n shares of the stock as in expression (20.8). This implies that if the stock is currently in-the-money, n = 1, and if it is out-of-the-money, n = 0. - If the call is out-of-the-money, then delta hedging using expression (20.8) is equivalent to not hedging! In this case, hedging the Greek based on expression (20.8) is useless. - If the call is in-the-money, then delta hedging using expression (20.8) is equivalent to shorting the entire stock. In this case, let us investigate the improvement, if any, in the hedging based on expression (20.8) versus doing nothing at all.

Hedging Based on Expression (20.8) Does Not Work ■

To prove that delta hedging based on the formula in expression (20.8) doesn’t work when the call is in-the-money, we need to know the true random evolution of the stock price.

■

Because we are now familiar with the BSM model, let’s assume that the stock price actually satisfies the BSM lognormal distribution assumption from Chapter 19. Then, the “true” price of the option is the BSM value, and we can approximate the change in the true option value using the BSM delta and a Taylor series expansion, that is, Δc =

𝜕c 𝜕c 𝜕c Δt + ΔS = Δt + N (d1 ) ΔS 𝜕t 𝜕S 𝜕t

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■

Suppose that we delta hedge using expression (20.8) with one share of the stock. By direct substitution, we see that change in the value of the delta-hedged portfolio is ΔV = Δc − ΔS =

𝜕c 𝜕c 𝜕c Δt + ΔS − ΔS = Δt + [N (d1 ) − 1] ΔS 𝜕t 𝜕S 𝜕t

(20.9)

If the delta hedge works, then the change in this portfolio’s value should be just (𝜕c/𝜕t)∆t. Why? Because this term is deterministic, and it would imply that var(∆V ) = 0. In contradiction, expression (20.9) shows that the hedging error of the hedged portfolio V is [N(d1 ) – 1] ∆S. ■

Now if we didn’t hedge at all, the change in the value of the portfolio would equal the change in the value of the option, that is, ΔV = Δc =

■

𝜕c 𝜕c 𝜕c Δt + ΔS = Δt + N (d1 ) ΔS 𝜕t 𝜕S 𝜕t

(20.10)

We can now see that delta hedging has really not improved our situation very much! - The hedging errors are [1 – N(d1 )] ∆S in expression (20.9) versus N(d1 )∆S in expression (20.10). - For just in-the-money options, N(d1 ) ≈ 0.6. So the delta-hedged portfolio change (expression [20.9]) and the naked call portfolio (expression [20.10]) are quite close because [1 – N(d1 )] ≈ 0.4 and N(d1 ) ≈ 0.6.

Summary

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■

Why does delta hedging work for the BSM model and not for this example? Because in the BSM model, the stock price is stochastic and its risk is priced within the model. Hedging its Greek is appropriate. In our example, however, the stock price is nonrandom, so its risk is not priced within the model. Hedging its Greek, the delta, using this call model doesn’t work.

Now let us turn to vega hedging in the BSM model. Just as in our previous example, because the BSM model from which the vega is obtained has no volatility risk included, volatility risk is not priced within the model. The Taylor series expansion is irrelevant because the volatility is constant (∆𝜎 = 0) in the derivation of the BSM model. In the BSM model, vega measures how the BSM’s price changes if the volatility is changed once and forever constant thereafter. This is not the case with stochastic volatility. Hence vega hedging is meaningless in this context and will not work. The magnitude of the error in vega hedging, like our example, is understood by looking at an OPM that relaxes the constant volatility assumption. These are called stochastic volatility option models. Extension 20.2 illustrates the bias in vega hedging and provides a brief look into stochastic option volatility models.

VEGA HEDGING

473

EXTENSION 20.1: Stochastic Volatility Option Models This extension shows the magnitude of the errors that occur in vega hedging when using the BSM formula. To see the bias, we consider a more general stochastic volatility OPM in which the stock’s volatility is stochastic. In this more general model, in which volatility risk is priced, we can compute the correct vega and compare it to the vega from the BSM model. The difference between the two vegas characterizes the error in using the BSM vega. For a stochastic volatility option pricing model, we need to change the constant volatility assumption A8 of Chapter 19. This is done by retaining the same structure as expression (19.2) of Chapter 19 but by modifying the volatility term from a constant 𝜎 to a new expression 𝜎(t), which is random. The stock price evolution is given by 𝜎(t)2 log S (t + Δt) = log S (t) + μ − Δt + 𝜎 (t) √Δtz (1) ( 2 ) where S(t + ∆t) is the stock price at time t + ∆t, μ is the expected return on the stock per year, z is a standard normal random variable with mean 0 and variance 1, and the evolution of the volatility is given by 𝜎 (t + Δt) = 𝜎 (t) + 𝜂 (t) Δt + 𝛽1 (t) √Δtz + 𝛽2 (t) √Δtw

(2)

where 𝜂(t) is the expected change in the volatility per year, 𝛽1 (t) and 𝛽2 (t) are volatilities of the volatilities, and w is another standard normal random variable with mean 0 and variance 1 that is independent of z. In this model, both the stock price and the stock’s volatility follow stochastic processes. It can be shown (see Eisenberg and Jarrow 1994) that the price of a European call option with strike price K and maturity T is

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∞

c (0) = ∫

BSM (𝜃0 ) 𝜋 (𝜃0 |𝜎0 ) d𝜃0

(3)

0

where BSM(𝜃0 ) is the BSM formula,

T

𝜃0 = ∫ 𝜎s ds 0

and the pseudo-probability of 𝜃0 conditional on 𝜎0 is 𝜋 (.|𝜎0 ) We see that the call option’s true price is a weighted average of BSM values. The correct hedge ratio is therefore 𝜕c (0) =∫ 𝜕𝜎0 0

∞

𝜕𝜋 (θ0 |𝜎0 ) 𝜕BSM (θ0 ) 𝜕θ0 𝜋 (θ0 |𝜎0 ) + BSM (𝜃0 ) dθ0 𝜕θ0 𝜕𝜎0 𝜕𝜎0 [ ]

(4)

Clearly this is different from the BSM vega, which is 𝜕BSM(𝜎0 )/ 𝜕𝜎0 in the notation of this extension. As such, the difference between these two vegas documents the error in using the BSM vega when the volatility is stochastic. The result is that hedging using the BSM vega when volatility is stochastic doesn’t work.

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20.5

Calibration

In the world of applied sciences and engineering, calibration is the process of setting a device’s measurements in tune with the measurements of another device with a known magnitude. For example, the unit of distance meter was measured from 1889 to 1960 in terms of a prototype “meter bar” kept in the International Bureau of Weights and Measures in France. To calibrate a second rod equal to one meter, one needs to compare it with the meter bar and mark off an identical distance. In derivatives pricing, calibration is a technique that estimates a derivatives pricing model’s parameters by equating the model’s price to the market price of a derivative. Before understanding the use and misuse of calibration, especially with respect to the BSM model, we need to refine our understanding of economic models. There are two types of models: theoretical and statistical.2

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Theoretical and Econometric Models ■

A theoretical model relies on assumptions regarding the structure of the economy and economic theory to derive a set of outputs (Figure 20.1 shows the basic structure of such models). The purpose is to understand cause and effect. For example, the BSM model is a theoretical model. In this case, we have discussed the assumptions underlying the model. Its parameters are estimated using historical stock price data, as described in Chapter 19, and the outputs are the option values and the Greeks (delta and gamma). The model is designed to understand the impact of the inputs (cause) on the outputs (effect).

■

A statistical model or, equivalently, an econometric model (because our model arises in an economic context) identifies patterns in historical data— correlations—that are expected to continue into the future. Econometric models use statistical techniques to model the patterns in historical data. Recall that more than a century ago, London dealers used a very simple statistical model to price options. A similar statistical model for pricing options can be generated as follows. Using historical option prices, compute the average call price conditioned on the strike price, the time to maturity, the stock’s volatility, and the level of the stock price. Use this conditional average call price as an estimate of the traded option’s value. Here the conditional averages are capturing the correlations between the inputs (strike, maturity, volatility, stock price) and the output (option value). A more sophisticated statistical model for pricing call options is discussed later, which uses implied volatilities. Both types of models have their advantages and disadvantages. Models are designed to understand the structural shifts in a dynamic and changing economy. Theoretical models hope to understand the casual relationships in these structural changes. If modeled correctly, a theoretical model is useful both before and after the structural shifts. The disadvantage of a theoretical model is that it is hard to

2

See pp. 2–3 of Applied Time Series Analysis f or Managerial Forecasting by Charles R. Nelson (1973) for a discussion of theoretical models (which have mathematical functions to represent “causal relationships” in some economic environment) vs. statistical (econometric) models.

CALIBRATION

FIGURE 20.1: Theoretical Vs. Econometric Models Theoretical Model (Cause and Effect): One validates a theoretical model by testing the consistency of the outputs and assumptions against real-world data. Assumptions

Real world

Model

Outputs

Validation Econometric/Statistical Model (Correlations): One validates a statistical model by testing the outputs against real-world data. Real world

Model

Outputs

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Validation

get it right. In contrast, statistical models are often easier to construct, but they frequently do not capture structural shifts. Consequently, if a structural shift occurs, the statistical model will be in error until it is reestimated. Statistical models are indispensable when no theoretical models can be built (we discussed an example of this in the context of risk minimization hedging in Chapter 13). Sometimes, as we illustrate later in this chapter, a theoretical model can degenerate into a statistical model.

Using Calibration Calibration has two uses with respect to derivatives pricing models. These two uses depend on whether the model is accepted or rejected when it is tested with respect to historical data.

DERIVATIVES PRICING MODEL ACCEPTED If the derivatives pricing model is accepted (consistent with historical data), then calibration can be used as a quick method for estimating the model’s parameters. Note that in this case, the estimated model’s parameters will be the same regardless of whether they are estimated using historical data or via calibration. However, this is

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not the usual circumstance under which calibration is used. The more usual circumstance is when the derivatives pricing model is rejected by historical data.

DERIVATIVES PRICING MODEL REJECTED If the derivatives pricing model is rejected (inconsistent with historical data), then calibration is used to make the model’s price consistent with the market price. Note that in this case, calibration is used to fix or overcome a pricing error embedded in the derivatives pricing model. The process of calibration thus transforms a theoretical model into an econometric model. This econometric model provides a nonlinear estimate of the derivative’s price. The nonlinear estimator results from the use of the inverse of the derivatives pricing formula. Given that calibration generates a statistical model formulated to match market prices, the derivatives model should be used for no other purpose.3 In this case, the calibrated model should only be used to price derivatives. The theoretical model, being rejected, can no longer be used to perform delta and gamma hedging. We will return to this comment later, when we discuss implied volatilities, but before this, we provide a simple example to illustrate these insights.

EXAMPLE 20.5: Calibration and Hedging Errors To illustrate the incorrectness of using a calibrated derivatives pricing model to hedge when the underlying model is rejected by the data, we return to Example 20.4, in which the stock price S(t) is non-random and grows at the risk-free rate r. This model is easily rejected using historical stock price data. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

Recall that under this assumption, the value of a European call option with strike price K and maturity T is c (t) = max [S (t) − Ke−r(T−t) , 0] (20.11) and the option’s details is 1 if S − Ke−r(T−t) > 0 𝜕c = 𝜕S {0 otherwise

(20.12)

If we were to use this model to price options, we would discover that the model price does not match the market price. This is not surprising. ■

Nonetheless, we can still use the model as a statistical model for pricing options if we use calibration. In this case, the parameter to calibrate is the stock price itself. We use calibration to find an implicit stock price, that is, that stock price S* that equates the model price to the market price: cMkt (t) = S∗ − Ke−r(T−t)

3

Thomas J. Sargent of New York University discusses similar problems in macroeconomic models; see “An Interview with Thomas J. Sargent by George W. Evans and Seppo Honkapohja” in Macroeconomic Dynamics 9 (2005).

CALIBRATION

Such an implied stock price S* always exists, and of course, the implied stock price will not equal the market price of the stock. However, this is not a concern because the purpose for using the model is to match options prices (not stock prices). Note the analogy of the implied stock price to an implied volatility in the context of the BSM model, in which volatility is held constant. ■

Interestingly, we can then use this statistical model with the implied stock price to value different strike and different maturity options, and it will fit them reasonably well. This is a valid use of the calibrated statistical model.

■

An improper use would be to use the statistical model, with the implied stock price, to hedge an option. The option’s delta is either 1 or 0, which as noted before provides a very poor hedge.

Implied Volatilities

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Computing an implied volatility with respect to the BSM model is the prime example of calibrating a derivatives pricing model. Hence we can apply our insights from the previous discussion of calibration to this situation. Before that, however, we recall the definition of the BSM implied volatility. Consider a European call option on a stock with strike price K, maturity T, continuously compounded risk-free rate r, and stock return volatility 𝜎. Denote the BSM model of chapter 19 by BSM(t, S, 𝜎, r, K, T). The implied volatility is that 𝜎IV such that the model call price matches the market price; that is, 𝜎IV is chosen to make the following equation hold: BSM (t, S, 𝜎IV , r, K, T) = cMkt

(20.13)

Thus, for implied volatility, we use (t, S, r, K, T) and cMKt as inputs and then compute 𝜎IV by solving this equation. Technically, since the BSM formula is an increasing function of the volatility, a solution always exists. Because the equation has no explicit analytical solution, numerical approximation techniques are used to compute the implied volatility 𝜎IV . We can rewrite the implied volatility in a more abstract form as 𝜎IV = BSM−1 (cMkt ∶ t, S, r, K, T)

(20.14)

where BSM−1 ( . ) denotes the inverse function. As indicated, this expression gives a nonlinear estimator for the option’s volatility as a function of the inputs (cMkt , t, S, r, K, T). Next, we illustrate the computation of an implied volatility.

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EXAMPLE 20.6: Computing Implied Volatilities ■

Table 20.3 reports closing option prices for International Business Machine Corp. (IBM) on February 3, 2011, along with relevant input data: - Column 1 reports six strike prices $155 to $175 at $5 intervals. - Column 2 reports the closing stock price on IBM. - Column 3 reports the continuously compounded risk-free interest rate r = 0.10 per year (chapter 2 shows how to compute r from Treasury bill prices). - Column 4 reports the time to maturity. Because options expire on February 19, there are sixteen days remaining before maturity. Thus, T = 16/365 = 0.0438 years. - Column 5 reports the call prices from the market. We assume that these options are European. - Column 6 reports the implied volatility. The implied volatility in column 6 are obtained in Excel by first guessing a value for the implied volatility, using this and the other inputs to determine cBSM , and then using Excel’s “Goal Seek” command to solve for the implied volatility that would set the difference (cMkt – cBSM ) to zero.4

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TABLE 20.3: Implied Volatilities K

S

r

T

cMkt

Implied Volatility

150

163.53

0.001

0.0438

13.575

0.1938

155

163.53

0.001

0.0438

8.6

0.1405

160

163.53

0.001

0.0438

4.0

0.1234

165

163.53

0.001

0.0438

1.08

0.1249

170

163.53

0.001

0.0438

0.17

0.1324

175

163.53

0.001

0.0438

0.055

0.1648

■

Figure 20.2 plots the strike prices (column 1) along the horizontal axis and implied volatilities corresponding to each strike along the vertical axis. This gives a pattern that is referred to as a volatility smile.

■

When computing implied volatilities, if the BSM model were consistent with the data, then for all strikes, one would get the same value. The graph would be a flat line. The implied volatilities are not equal, hence the BSM model is rejected by this example’s data.

CALIBRATION

FIGURE 20.2: Implied Volatility Smiles Implied volatility 0.25 0.20 0.15 0.10 0.05 0.00 150

155

160

165

170

175 Strike Prices

4

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You can solve for implied volatility by (1) trial and error, (2) by a spreadsheet program as outlined above (see the problems at the end of this chapter for more details), or (3) by using our Priced! software.

To understand the proper use of the calibrated BSM with an implied volatility, we first need to determine if the BSM model is accepted or rejected by historical data. Unfortunately, it is well known that the BSM model is rejected by historical data. The most direct verification of this rejection has been studies computing implied volatilities (as in the previous example). When computing implied volatilities, if the BSM model were correct, then for all the other inputs (t, S, K, r, T), one would get the same value for the volatility, and it would be equal to the historical volatility. Unfortunately, when estimating implied volatilities from equity options, the evidence shows that (1) implied volatilities differ across strikes and maturities (the pattern is a smile or sneer; see Figure 20.2 for a typical pattern); (2) implied volatilities differ across time, stock prices, and interest rates; and (3) implied volatilities do not equal historical volatilities (see Bakshi et al. 1997; Pan 2002). This evidence strongly rejects the BSM model’s validity. Given that the BSM model is rejected, one can still use a calibrated BSM as a nonlinear statistical model for estimating options prices. This is typical industry practice. A common procedure is to estimate the implied volatility using a subset of the available options and then to use the resulting implied volatility estimate to price the remaining set of options or options at a later date. This is a perfectly valid use of the BSM model.

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However, because the BSM model is rejected by historical data, it is incorrect to use a calibrated BSM to delta or gamma hedge. This is a common misuse of the BSM model. The hedging implications of the theoretical model are no longer valid. To hedge in this circumstance, one needs to employ either an alternative and more complex theoretical model or a purely statistical model (as was done for futures prices in Chapter 13). Vega hedging will not work, contrary to common belief, for two reasons: first, it is a wrong Greek to hedge because it was held constant in the original BSM model (volatility risk is not reflected in the BSM price), and second, the BSM model is rejected by historical data, hence its partial derivatives—the Greeks—are irrelevant. For a study showing the poor hedging that results from using the BSM model with delta, gamma, and vega hedging, see Engle and Rosenberg (2000).

EXTENSION 20.2: VIX: The Fear Index If you think implied volatility is scary, it has been used to develop something even scarier—the “fear index,” called the VIX. The Chicago Board Options Exchange introduced the VIX in 1993 “to measure the market’s expectation of the 30-day volatility implied by at-the-money S&P 100 Index (OEX) option prices.”5 In 2003, the VIX index was modified to represent an expected volatility instead of an implied volatility. Today, the VIX has developed into a widely watched barometer of stock market volatility, reflecting investor anxiety. The VIX index increases during financial crises. Today, various derivatives trade on VIX, yielding investors and financial institutions a method to hedge or speculate on stock market volatilities

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5

See “The CBOE Volatility Index – VIX” (www.cboe.com/micro/vix/vixwhite.pdf), page 2.

20.6

The Black–Scholes–Merton Model: A Postscript

As shown, because the BSM model is rejected by historical data, although a calibrated BSM model is still useful for pricing options, it is not useful for hedging. A calibrated BSM model is a statistical model in which the implied volatility estimate captures any model errors into its magnitude. These model errors, however, invalidate its use for hedging. So what can one do to overcome the hedging limitation of the BSM model? There are two approaches. First, one can build a statistical model relating changes in the option’s price to changes in the stock’s price, analogous to that used in Chapter 13 for futures prices. Such a statistical model will inherit all the advantages and disadvantages of such models in predicting the future. Second, one can build a better mousetrap! The better mousetrap would generalize the assumptions so that the option model is no longer rejected by historical data. A preliminary guide to these extensions is as follows. Extension 20.2 discusses how to extend the BSM model to include volatility risk. Chapters 21–25 of this

SUMMARY

book discuss how to include interest rate risk in OPMs. Asset price bubbles and their impact on option pricing were discussed in Chapter 19. Finally, credit and liquidity risk extensions of OPMs are discussed in Chapter 26. Transaction costs and trading restrictions are also worthy of study, but we leave their discussion to more advanced textbooks and the academic literature.

20.7

Summary

1. Delta hedging is removing the price risk from a long call position by selling short deltaC ≡ where d1 =

𝜕c = N (d1 ) shares of stock 𝜕S

𝜎2 S T log ( ) + r + ( K 2) σ√T

A delta hedge works well when the time interval over which the hedge is executed is small.

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2. Delta and gamma hedging removes the price risk from a long call with price c1 by buying n1 shares of stock with price S and selling n2 shares of another option on the same stock with price c2 , where n1 = (gammaC1 /gammaC2 ) deltaC2 − deltaC1 and n2 = −gammaC1 /gammaC2 The Greeks are defined by the general formulas deltaC ≡

𝜕c 𝜕2 c 1 N′(d1 ) = N (d1 ) and gammaC ≡ 2 = 𝜕S 𝜕S Sσ√T

where N( . ) is the cumulative standard normal distribution function and N ′ ( . ) is the standard normal density function. Delta and gamma hedging are designed to overcome the errors introduced when the hedging interval is not small. 3. Vega hedging is a nonsensical procedure because it is based on an option valuation formula, the BSM model, that assumes volatility is a constant. Hence volatility risk is not embedded in the BSM formula, so the formula’s sensitivity to changes in the stock’s volatility, the vega, does not capture volatility risk in an option’s price. 4. Calibration is an often-misused estimation procedure. In the context of the BSM formula, implied volatility estimation is an example of calibration. Because the BSM is rejected by historical data, using a calibrated BSM model transforms the theoretical BSM model into a statistical model. This statistical model can be used to estimate options prices, but it is invalid to use a calibrated BSM model to delta (and/or gamma) hedge an option position.

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20.8

Appendix

The Mathematics of Delta, Gamma, and Vega Hedging Hedging using the Greeks is based on a Taylor series expansion. A Taylor series expresses a well-behaved function as a sum of terms involving its derivatives computed at a single point. For a function f of two variables x and y, the Taylor series about the point (a, b) can be written as (where [x – a] is ∆x, [y – b] is ∆y, f [x, y] – f [a, b] is ∆f ; see Appendix A) Δf =

𝜕2 f 𝜕f 𝜕f 1 𝜕2 f 2 𝜕2 f 2 Δy + 2 Δx + Δy + Δx + ΔxΔy + error ] 2 [ 𝜕x2 𝜕x 𝜕y 𝜕x𝜕y 𝜕y2

where the higher-order terms of the expansion are considered insignificant “error terms” when ∆x and ∆y are small.

DELTA HEDGING Suppose one can trade quickly enough so that the infinitesimal time change dt is approximately equal to the small time change ∆t that we consider in practice, that is, dt ≈ ∆t. Then the BSM theory implies that delta hedging removes all of the stock price risk from an option position. We now explain how this happens.

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■

The option and the stock portfolio. Consider a portfolio consisting of one European call option with time t price c(t). We want to trade n shares of a stock worth S(t) to remove the stock price risk from this option position. Denote the time t value of the portfolio as V (t). In symbols, we can write the evolution of the change in the portfolio’s value over a small time interval [t, t + ∆t] as ΔV = Δc + nΔS

(1)

where ∆V ≡ V (t + ∆t) – V (t) and the other terms are similarly defined. ■

The Taylor series expansion for a delta hedge. Because the BSM call price is a function of time t and the stock price S(t) (other terms like the volatility, the interest rate, and the strike price are constant parameters and not variables), we can use a Taylor series expansion to write Δc =

𝜕c 𝜕c Δt + ΔS + error 𝜕t 𝜕S

(2)

where (error/∆t) goes to zero as the time interval shrinks to zero (∆t → 0). This last observation implies that for small changes in time, ∆t, the error term is small and can be ignored. Ignoring these terms, substitution of expression (2) into expression (1) gives the change in the value of our portfolio as ΔV =

𝜕c 𝜕c Δt + ΔS + nΔS 𝜕t 𝜕S

(3)

APPENDIX

■

Determining the hedge ratio and constructing the delta-neutral portfolio. To make this portfolio riskless for small changes in the stock price, choose n so that the term involving ∆S disappears. This is the only random term in expression (3). This is obtained by setting 𝜕c ΔS + nΔS = 0 (4) 𝜕S Solving for n yields 𝜕c n=− = −N (d1 ) ≡ −deltaC (5) 𝜕S The number of shares held has the opposite sign to the option position. Because we began with one long call, we need to short deltaC ≡ N(d1 ) shares of the underlying stock to hedge the exposure.

■

The delta is the holy grail of options pricing. The hedge ratio n represents the number of shares of the stock to trade for each call to construct a riskless portfolio. Being riskless, this portfolio is equivalent to a position in a money market account. To see this, substituting n into expression (3) gives

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ΔV =

𝜕c Δt 𝜕t

(6)

Thus the change in the portfolio’s value equals 𝜕c/𝜕t (which equals the call’s theta; see Table 19.1) times the time interval ∆t. This term is nonrandom, and hence it has zero variance. To avoid arbitrage, this portfolio must earn the riskless rate. This observation (combined with an extension of this approach, as described in the next section in the context of gamma hedging) leads to an alternative derivation of the BSM model. The portfolio V that we have constructed is said to be delta neutral. Notice that delta hedging involves the first-order derivatives. Such approximations work well over small time periods, during which the stock price does not move much.

GAMMA HEDGING How does one remove the price risk of an option position over a time interval that is longer than what we had considered earlier: a day, or perhaps even a week (in other words, the hedging interval ∆t is significantly larger than the infinitesimal time interval dt considered in the model)? We can achieve this by adding a gamma hedge to the delta hedge. ■

The Taylor series expansion for delta and gamma hedging. Suppose that we are long one European call option and we want to remove the stock price risk from our portfolio over the time interval [t, t + ∆t]. As before, we start with a Taylor series expansion of the call’s value, but this time, we include the second-order term: Δc =

𝜕c 1 𝜕2 c 𝜕c (ΔS)2 + error Δt + ΔS + 2 𝜕S2 𝜕t 𝜕S

(7)

where (error/∆t) goes to zero as ∆t→ 0. In the calculation, we ignore the error term because it is small.

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■

Adding a stock and another option to the option portfolio. Notice that there are “two” risks in the call’s price movements—one comes from the change in the stock price and the other from the change in the stock price squared. To hedge two risks, we need two securities. We can use the underlying stock as one security and another option on the same stock as the second. At time t, we create a portfolio V (t) consisting of one long call whose price is c1 , n1 shares of the stock whose price is S(t), and n2 number of the second option whose price is c2 : V (t) = c1 + n1 S + n2 c2

■

(8a)

Constructing the delta- and gamma-neutral portfolio. To determine the portfolio’s composition, we write its value change over [t, t + ∆t] as ΔV (t) = Δc1 + n1 ΔS + n2 Δc2

(8b)

- Substitution of expression (7) into (8b) for the change in the two calls’ values gives

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ΔV =

𝜕c 𝜕c1 1 𝜕2 c1 Δt + 1 ΔS + (Δs)2 + n1 ΔS 2 𝜕S2 𝜕t 𝜕S 𝜕c 𝜕c2 1 𝜕2 c2 (ΔS)2 +n2 2 Δt + n2 ΔS + n2 2 𝜕S2 𝜕t 𝜕S

(9)

- To make this portfolio have zero risk, we want to choose n1 and n2 such that the coefficients of the terms involving ∆S and (∆S)2 disappear. This gives two equations in two unknowns: 𝜕c 𝜕c1 + n1 + n2 2 = 0 𝜕S 𝜕S

(10a)

𝜕2 c2 𝜕2 c1 + n =0 (10b) 2 𝜕S2 𝜕S2 Notice that expression (10a) involves the Greek deltas and that the second expression (10b) involve the Greek gammas (see Table 19.1). - Rearrange terms in (10b) to get n2 = −

𝜕2 c1 𝜕2 c2 / ( 𝜕S2 ) ( 𝜕S2 )

(11a)

The negative sign indicates that we need to sell short n2 options. Alternatively, you can write n2 as –gammaC1 /gammaC2 .

APPENDIX

- Plug this value into (10a) and rearrange terms to get n1 =

𝜕c2 𝜕c1 𝜕2 c1 𝜕2 c2 − / [( 𝜕S2 ) ( 𝜕S2 )] 𝜕S 𝜕S

(11b)

This can also be written as (gammaC1 /gammaC2 )deltaC2 – deltaC1 . The hedge involves going long n1 shares of the stock if the preceding expression is positive and selling them short if the expression is negative. - Substituting these shares into the portfolio’s value shows that the resulting portfolio is riskless: ΔV =

𝜕c2 𝜕2 c1 𝜕2 c2 𝜕c1 Δt − / Δt 2 2 [( ) ( )] 𝜕t 𝜕t 𝜕S 𝜕S

(12)

Such a portfolio, which consists of nonrandom terms, is said to be delta- and gamma-neutral.

VEGA HEDGING As we discuss in the text, vega hedging is “the bad and the ugly” and should not be used. To see why this is an invalid procedure, let us begin by following the logic that we employed for both delta and gamma hedging.

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- The Taylor series expansion for a delta and vega hedge. We start with the BSM formula, which depends on the stock price, time, and volatility, written c[S(t), t, 𝜎]. To hedge both the delta and vega over [t, t + ∆t], we proceed to use a Taylor series expansion, as before: Δc =

𝜕c 𝜕c 𝜕c Δt + ΔS + Δσ + error 𝜕t 𝜕S 𝜕𝜎

(13)

where (error/∆t) goes to zero as ∆t → 0. Because the error term is small, we ignore it in the subsequent calculations. - Adding a stock and another option to the portfolio. There are two risks in the call’s price movements: one is the change in the stock price and the other is the change in the stock’s volatility. As before, we hedge them using the underlying stock and a second option on the same stock. This gives the portfolio’s value at time t: V (t) = c1 + n1 S + n2 c2

(14)

- Constructing the delta-and vega-neutral portfolio. We write the change in the portfolio’s value over [t, t + ∆t] as ΔV (t) = Δc1 + n1 ΔS + n2 Δc2

(15)

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Substitution of the change in the call’s value gives an expression analogous to expression (9): ΔV =

𝜕c1 𝜕c 𝜕c Δt + 1 ΔS + 1 Δ𝜎 + n1 ΔS 𝜕t 𝜕S 𝜕𝜎 𝜕c2 𝜕c2 𝜕c +n2 Δt + n2 ΔS + n2 2 Δ𝜎 𝜕t 𝜕S 𝜕𝜎

(16)

- To make this portfolio have zero risk, we choose n1 and n2 such that the terms involving ∆S and ∆𝜎 disappear. This is accomplished by requiring 𝜕c 𝜕c1 + n1 + n2 2 = 0 𝜕S 𝜕S 𝜕c1 𝜕c + n2 2 = 0 𝜕𝜎 𝜕𝜎 - Rearrange terms in (17b) to get n2 = −

𝜕c2 𝜕c1 / ( 𝜕𝜎 ) ( 𝜕𝜎 )

(17a) (17b)

(18a)

A negative sign indicates that we need to sell short vegaC1 /vegaC2 options. - Plugging this value into expression (17a) and rearranging terms yields

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n1 = ■

𝜕c2 𝜕c1 𝜕c2 𝜕c1 / − [( 𝜕𝜎 ) ( 𝜕𝜎 )] 𝜕S 𝜕S

(18b)

However, there is a conceptual problem in this application. We have kept the parameter 𝜎 fixed throughout our derivation of the BSM model, and now we want to change it! This is nonsensical because the risk of a changing volatility is not reflected in the option’s value. Consequently, vega hedging is an invalid procedure that should never be used.

A Third Derivation of the Black–Scholes–Merton Formula The original derivation of the BSM formula in 1973 for valuing a European option was obtained as the solution to a partial differential equation. It follows from the Taylor series expansion used for delta and gamma hedging, with one small twist. Of course, this approach needs the underlying assumptions given in chapter 19. Assume that the European call option’s value depends only on the price of the stock and time, that is, c(t) = c[t, S(t)]. Then, you can create a “hedged position” by combining a long position in the stock with a short position in the option. Letting the short position in the stock be (𝜕c/𝜕S) shares, the portfolio’s value is V=c−

𝜕c S 𝜕S

(19)

CASES

■

■

Suppose that the stock price changes by ∆S ≡ S(t + ∆t) – S(t) over a small interval [t, t + ∆t] and the call price changes by ∆c. Then, the change in the value of our portfolio is 𝜕c ΔV = Δc = ΔS (20) 𝜕S Use of Ito’s lemma and stochastic calculus. We can use stochastic calculus to develop an alternate expression for the change in the portfolio’s value: - If the option position changes continuously, Ito’s lemma yields Δc =

𝜕c 𝜕c 1 𝜕2 c Δt + ΔS + (ΔS)2 2 𝜕S2 𝜕t 𝜕S

(21)

This is similar to equation (7). - Expression (18) can be further simplified by using another result from stochastic calculus, which allows us to write the squared change in the stock price as (∆S)2 = 𝜎2 S2 ∆t. This is the “one small twist” mentioned earlier. - Substituting into (21) and applying it to (20) yields 𝜕c 1 𝜕2 c 2 2 𝜕c 𝜕c Δt + ΔS + 𝜎 S Δt − ΔS 2 2 𝜕S 𝜕t 𝜕S 𝜕S 2 𝜕c 1𝜕c 2 2 = Δt + 𝜎 S Δt 2 𝜕S2 𝜕t

ΔV =

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■

■

A riskless hedge. Because all terms on the right side of (22) are deterministic, no arbitrage necessitates that the hedged portfolio V must earn the risk-free rate over the time period ∆t, that is, rV ∆t. Substitution and canceling ∆t from both sides gives 𝜕c 1 𝜕2 c 2 2 𝜎S (23) rV = + 𝜕t 2 𝜕S2 Black–Scholes–Merton partial differential equation. Substituting (19) into (23) and rearranging terms gives the famous BSM partial differential equation (pde) 𝜕c 1 𝜕c 𝜕2 c = rc − rS − 𝜎2 S2 2 𝜕t 𝜕S 2 𝜕S

■

(22)

(24)

Derivation of the Black–Scholes–Merton formula. Using the call option’s boundary condition c(T) = max[S(T) – K, 0], the solution to this pde is the BSM formula.

20.9

Cases

J&L Railroad (Darden School of Business Case UV0251-PDF-ENG, Harvard Busi-

ness Publishing) and J&L Railroad: The Board Meeting (Darden School of Business Case UV2566-PDF-ENG, Harvard Business Publishing). The first case studies a railroad company’s decision to hedge next year’s expected fuel demand.

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The second case, which may be done in conjunction with the first case, considers these issues in greater detail. Pine Street Capital (Harvard Business School Case 201071-PDF-ENG). The case

examines issues faced by a technology hedge fund in trying to decide whether and/or how to hedge equity market risk by short selling and trading options. Sleepless in LA (Richard Ivey School of Business Foundation Case 905N11-PDF-

ENG, Harvard Business Publishing). The case discusses the Black–Scholes– Merton model for options pricing, the concept of implied volatility, and put–call parity. It also shows how options pricing can be used to value the corporate liabilities of a financially distressed company.

20.10 Questions and Problems The first five questions use the following information for pricing European options. Special Motors Corporation’s stock price S is $59, the strike price K is $60, the maturity T is forty-four days, the implied volatility 𝜎 is 30 percent per year, and the risk-free interest rate r is 3.3 percent per year. 20.1 a. Given the preceding data, compute the call option’s delta using the Black–

Scholes–Merton model. b. Given the preceding data, compute the put option’s delta using the Black–

Scholes–Merton model. 20.2 a. Describe how to set up the delta hedge for a long call option position based Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

on the computations in problem 20.3. b. If the stock price changes to $60 over the next trading day, what is the

delta hedging error? Assume the Black–Scholes–Merton model accurately describes movements in the call’s value. 20.3 a. Describe how you will set up the delta hedge for a short put option position

based on computations in problem 20.3. b. If the stock price changes to $60 over the next trading day, what is the

delta hedging error? Assume the Black–Scholes–Merton model accurately describes movements in the call’s value. 20.4 a. Given the data in question 20.3, compute the call option’s gamma on the

basis of the Black–Scholes–Merton model. b. Given the data in question 20.3, compute the put option’s gamma on the

basis of the Black–Scholes–Merton model. 20.5 a. Consider a portfolio consisting of 50 shares of the stock, long 10 calls and

short 30 puts. What is the portfolio’s delta? (Use delta values from question 20.3.)

QUESTIONS AND PROBLEMS

20.6 a. What is vega hedging? b. Describe some conceptual problems with this once-popular practice. 20.7 If you want to hedge volatility risk in a call option position, how would you

go about doing so? 20.8 What is calibration with respect to the BSM model? Explain how to use

calibration correctly. When is calibration used incorrectly? Implied Volatilities K

cMkt

150

13.00

155

8.00

160

3.75

165

1.00

170

0.20

175

0.05

Implied Volatility

20.9 Briefly describe the two types of models that we introduced, highlighting their

strengths and weaknesses.

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20.10 When constructing a theoretical model for pricing a derivative, if the theo-

retical model is rejected by historical data, can one still use the model to price derivatives? Explain your answer using the BSM as an example. 20.11 Complete the following table using the following information:

Comment on what this implies about the validity of the Black–Scholes– Merton model? - Column 1 reports six strike prices $155 to $175 at $5 intervals. - Column 2 is for the closing stock price on YBM, $160. - Column 3 is for the risk-free interest rate r = 10 percent per year. - Column 4 is for the time to maturity, which is twenty days (20/265 = 0.054795 years). - Column 5 reports European call option prices from the market. - Column 6 reports the implied volatility that you need to calculate. 20.12 a. When using historical prices, is the Black–Scholes–Merton model accepted

or rejected by the data? b. If the model is accepted, explain why. If the model is rejected, give one

probable cause of rejection.

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20.13 If historical data rejects the Black–Scholes–Merton model, a. can the model still be used for pricing options? Explain your answer. b. can the model be used for hedging option positions? Explain your answer. 20.14 What is a volatility smile with respect to the Black–Scholes–Merton model? 20.15 a. What is delta hedging? If the Black–Scholes–Merton model is correct, do

you need delta hedging? b. What is gamma hedging? Under what conditions do you need gamma

hedging on top of delta hedging? 20.16 In hedging using the Black–Scholes–Merton model, we used the phrase “the

good, the bad, and the ugly.” Identify “the good,” and “the bad and the ugly” and explain why these descriptions are accurate or may be inappropriate. 20.17 Given the evidence supports that the stock’s volatility is stochastic and the

BSM model is rejected by the data, how can one hedge options? 20.18 (Microsoft Excel Project) We thank Professor Craig W. Holden of Indiana

University, Bloomington, for providing a similar problem (copyright-free) on his website (www. kelley. iu. edu/cholden). His website www. excelmodeling. com has several books teaching hands-on practical finance with the help of Microsoft Excel.

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Overview The purpose of this assignment is to introduce you to the science of pricing options. The project has two parts: ■

You will calculate and graph the value of European-style calls and puts using the built-in spreadsheet functions, such as the cumulative normal.

■

Using actual options data from the Wall Street Journal, you will also calculate the implied volatilities using the solver module and graph the “smile” pattern of implied volatility.

Data (from the Wall Street Journal, March 3, 2000) The closing price for the Standard and Poor’s 500 (S&P 500) index (symbol SPX) was 1,381.76. As options expire on the third Friday of the expiration month (17th for March contracts, 21st for April contracts), choose the (annualized) ask yield rate for the Treasury bill that matures nearest to the expiration date. Express T as fraction of a year: 14/365 for March options and 49/365 for April options.

QUESTIONS AND PROBLEMS

S&P 500 Index Put Option Prices Strike

Last (Price)

Strike

Last (Price)

Strike

Last (Price)

Mar 1325c

67

Mar 1375c

27

Mar 1425c

6 1/4

Mar 1325p

5 1/2

Mar 1375p

17

Mar 1425p

47 1/2

Apr 1325p

18 3/8

Apr 1375c

51

Apr 1425c

24

Mar 1350c

44

Apr 1375p

32

Apr 1425p

54

Mar 1350p

10

Mar 1400c

13 7/8

Mar 1450c

2 1/4

Apr 1350c

62 1/2

Mar 1400p

29

Mar 1450p

67

Apr 1350p

24

Apr 1400c

35 1/4

Apr 1450c

14 1/2

Apr 1400p

44

Apr 1450p

72 1/2

A “c” denotes call price and a “p” denotes put price. T-Bill Prices

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Maturity

Days to Maturity

Ask Yield

Mar 16

13

5.28

Mar 23

20

5.13

Mar 30

27

5.30

Apr 06

34

5.31

Apr 13

41

5.48

Apr 20

48

5.61

Apr 27

55

5.76

The Project Do the project in Excel worksheets. Print out and turn in all of your results. Deliverables are noted below. a. Cumulative Normal and Normal Density Graphs Examine a standard normal distribution. Find the cumulative normal and the normal density for values of d from –3.0 to 3.0 in increments of 0.1. Record this in a table in your Excel worksheet. Then graph both the cumulative normal and the normal density on the same diagram with d on the x axis.

491

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CHAPTER 20: USING THE BLACK–SCHOLES–MERTON MODEL

Deliverable: (1) The table with cumulative normal distribution function and the standard normal density function’s values. (2) The graph generated. b. Option Pricing Using Black–Scholes–Merton Formula and Intrinsic Value Computation Use the Black–Scholes–Merton formula to estimate the call price and the put price for options with the following data: Strike price K= 10 Risk-free interest rate r = 3 percent or 0.03 per year Volatility 𝜎 = 60 percent or 0.6 per year Time to maturity T = 1 year S(0) ranging from 0 (actually 0.0000001, a very small value) to 20, in increments of 2 Record these numbers and computations in a table. Deliverable: (3) Table with various inputs and corresponding call and put prices estimated using the BSM model. c. Intrinsic Value Computation and Graph Now calculate call intrinsic values and put intrinsic values for the same options (K = 10 and S(0) ranging from 0 to 20 in increments of 2).

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Graph both the call price and the call intrinsic value on the same diagram with S(0) on the x axis. Similarly, graph both the put price and the put intrinsic value on the same diagram with S(0) on the x axis. Deliverable: (4) Table with the call price and the call intrinsic value, and the put price and the put intrinsic value. (You may combine (3) and (4) and present it in a single table.) (5) The call graph. (6) The put graph. d. Implied Volatility Computation and Graph Given data from the March 3, 2000, issue of the Wall Street Journal, calculate the implied volatility from S&P 500 index put options (not index call options) for K (strike) values of 1325, 1350, 1375, 1400, and 1425 for the March contract. Then calculate the implied volatility from S&P 500 index put options (not index call options) for K values of 1325, 1350, 1375, 1400, and 1425 for the April contract. Graph the implied volatility diagram with the exercise price X on the x axis and the March and April implied volatilities on the y axis. Deliverable: (7) The table with the implied volatilities.

QUESTIONS AND PROBLEMS

(8) The graph with the implied volatilities. ( Suggestions and useful information, relevant excel commands, and implied volatility computation) Goal

Excel Command

1. To calculate the natural log:

=LN(A1)

2. To calculate the normal density:

=NORMDIST(A1,0,1,FALSE)

3. To calculate the cumulative normal:

=NORMDIST(A1,0,1,TRUE)

4. To calculate the exponential:

=EXP(A1)

5. To solve implied volatility:

Goal Seek

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To estimate implied volatilities using Goal Seek command in Excel 2007 or later versions: ■

In an Excel sheet, record different strike prices for March in one column and the various model inputs in other columns.

■

In the column with the heading “Implied volatility,” start with an arbitrary volatility number (say 0.2) and use it to compute March put price.

■

Record market-traded March put prices corresponding to the different strike prices. In another column, note the “Put Price Difference” to record the difference between “market traded put price” and the “model generated put price” solved from the model.

■

Next, select a cell under “Put Price Difference.” On the Data tab, in the Data Tools group, click What-If Analysis, and then click Goal Seek. - In the “Set cell” box, record the current cell box. - In the “To value” box, type in 0. - In the “By changing cell” box, type in the location of the cell corresponding to the implied volatility from the same row. - Click OK to get the implied volatility in its own column. (In this case, the zeroprice difference is obtained by changing the standard deviation.)

■

Hold and drag the mouse down to similarly solve implied volatilities corresponding to March and April put options corresponding to different strike prices.

493

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IV

Interest Rate Derivatives

CHAPTER 21

Yields and Forward Rates

CHAPTER 22

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Interest Rate Swaps

CHAPTER 23

CHAPTER 24

Single-Period Binomial HJM Model

Multiperiod Binomial HJM Model

CHAPTER 25

The HJM Libor Model

CHAPTER 26

Risk Risk Management Management Models

21 Yields and Forward Rates 21.1 Introduction EXTENSION 21.1 Home Ownership and Derivatives

21.2 Yields Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Revisiting Zeros Yields and the Yield Curve EXTENSION 21.2 The Expectations Hypothesis

21.3 The Traditional Approach Duration Modified Duration Hedging Applications and Limitations

21.4 Forward Rates The Definition Understanding Forward Rates

Using Forward Rates EXTENSION 21.3 Computing Forward Rates from Coupon Bond Prices

21.5 The Basic Interest Rate Derivatives Contracts A Brief History Forward Rate Agreements Interest Rate Futures EXTENSION 21.4 Treasury Futures The Equivalence between Forward and FRA Rates

21.6 Summary 21.7 Cases 21.8 Questions and Problems

INTRODUCTION

21.1

Introduction

In 1909, the founder of Ford Motor Company, Henry Ford, announced that the company would build only one car, the Model T, and famously declared, “Any customer can have a car painted any colour that he wants so long as it is black.”1 Similarly, mortgage loans used to be simple as well. In the 1950s and 1960s, home mortgages were typically made by savings and loans (S&Ls), whose standard offering was a no-frill 30-year, fixed-rate loan. S&Ls were known as 3-6-3s because they paid depositors 3 percent, charged borrowers 6 percent, and the managers went to play golf at 3:00 pm! But times have changed. Today home mortgages come with a variety of embedded options and complicated provisions (see Extension 21.1). There is a bewildering assortment of fixed-and adjustable- rate mortgages. Some have teaser rates with ceilings and floors on interest rates charged. Some don’t pay principal in the early years, whereas others are extendible. Most can be repaid without penalty. To understand these provisions— the embedded options—one needs to understand interest rate derivatives (IRDs). This is the purpose of this and the remaining chapters. Who uses IRDs? All players in the economy: individuals, corporations, governments, and financial institutions:

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1. Individuals. A family’s assets and liabilities are all sensitive to interest rate movements. On the liability side, interest rates affect payments on home mortgages, car loans, personal loans, and credit card balances. On the asset side, they affect earnings on savings accounts, certificates of deposit, and money market funds. 2. Corporations and governments. Corporations routinely use IRDs to structure their interest rate–sensitive liabilities and assets. Mutual funds, hedge funds, pension funds, insurance companies, and college and university endowments use IRDs to manage their investments. Last, they are essential for managing the complex treasury operations of local, state, and federal government. 3. Financial institutions. Financial institutions use advanced knowledge of IRDs to develop products for their clients and to manage their own interest rate risks. Our discussion of IRDs parallels our earlier presentation of spots, forwards, futures, and options on commodities and equities; however, there’s a twist. For IRDs, an array of different maturity default-free zero-coupon bonds replace the single commodity or stock price as the underlying asset. This small change raises some major hurdles for IRD pricing models. Still, despite its reputation for complexity, this subject is easy to understand if it is introduced gradually and with forethought. Magically, and perhaps without your awareness, the early chapters have already built this foundation for you! For simplicity, we focus on Eurodollar IRDs. They are simpler than derivatives on Treasury securities. Moreover, Eurodollar IRDs attract the greatest trading volume,

1

See Chapter IV of My Life and Work (1922) by Henry Ford in collaboration with Samuel Crowther from Project Gutenberg website www.gutenberg.org/dirs/etext05/hnfrd10.txt, February 23, 2011.

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CHAPTER 21: YIELDS AND FORWARD RATES

and you can have your cake and eat it too. Once you understand Eurodollar IRDs, it is easy to comprehend derivatives based on Treasuries.

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EXTENSION 21.1: Home Ownership and Derivatives Suppose you find your dream house and submit an offer to buy the house for $500,000. A flurry of activity now begins. If the seller accepts the offer, the two of you first create an option contract. The seller’s broker requires you to place a deposit, say, $5,000, with them. Should you decide to back out, the $5,000 will be forfeited; otherwise, it goes toward the purchase price. This deposit creates a call option to buy the house. You now go to a bank or a mortgage broker to arrange a loan. They obtain your credit score indicating how you have handled debt in the past: credit card as well as other loan payments. Your credit score helps determine whether you get the loan plus the magnitudes of the down payment required and the interest rate charged. On the basis of this evaluation, suppose that the lender agrees to loan you $450,000 for the house. This loan is called a mortgage. You have to pay this principal back in fifteen or thirty years with interest paid in monthly installments, and you have to come up with a $50,000 down payment. This 10 percent down payment provides a cushion against a decline in the house’s value if you decide to leave town without a forwarding address! Unless you make a 20 percent down payment, the lender sometimes requires you to pay for mortgage insurance, which is a put option on the risk of your not repaying the principal (a credit derivative!). Mortgages come in two varieties. A fixed-rate mortgage (FRM) charges a fixed interest rate over the loan’s life. For a FRM, you have to pay back the loan in equal monthly installments over fifteen or thirty years. An adjustable-rate mortgage (ARM) usually charges a lower initial interest rate for one to five years, with the remaining payments adjusted depending on future interest rates. ARM mortgages come with caps and floors on the interest paid per period to protect both the lender and the borrower from adverse interest rate movements. These caps and floors are call and put options on interest rates—more IRDs! Your expertise in modern finance should come in handy. If you are planning to stay in your home for a short period, or if you are betting that interest rates will remain low, or if you are anticipating a significant increase in your personal wealth in the near future, then an ARM may make sense. In addition, in the United States, most loans come without a prepayment penalty, so you can choose to pay back the loan before it matures. If interest rates fall, you also have the option of refinancing your home and locking in a cheaper interest rate. In years past, the major players in the mortgage market were banks and savings and loan associations (S&Ls). The money for mortgage loans was raised and lent in local markets. Thanks to modern finance, a mortgage broker can now lend you the money for a house mortgage and immediately sell that loan in a national market. Typically, a loan is split into two parts. The right to service the loan is held by entities like the Bank of America or JP Morgan Chase, which regularly collects your mortgage payments and coaxes you to buy other products. The loan itself may be combined with other mortgage loans and sold to Wall Street wizards, who then combine these mortgage loans into a pool and sell bonds against the pool to cover the cost of their purchases. These bonds are called asset-backed securities or, in this case, mortgage-backed securities. Did you really think the mortgage business was staid and stodgy?

On our journey to understanding IRDs, this chapter introduces yields and forward rates, which are fundamental concepts. We start by revisiting zero-coupon bonds, from which we extract yields, and the yield curve, concepts that are often introduced in basic finance and investment courses. We also discuss the once-popular but now

YIELDS

fading techniques of duration and convexity based interest rate risk management. Then, we introduce forward rates, which are at the heart of modern interest rate risk management. We wrap up this chapter with a discussion of forward rate agreements (FRAs) and interest rate futures, which are analogous to forwards and futures on commodities and equities.

21.2

Yields

To understand IRDS, we need to start at the beginning and understand yields. Yields are obtained from zero-coupon bond prices.

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Revisiting Zeros Our basic building blocks are risk-free zero-coupon bonds of different maturities. US Treasury bills and STRIPS and Eurodollar deposits fit our description of zeros. We introduced Treasuries and Eurodollars in Chapter 2. Although derivatives on the Treasuries ruled the markets during the early days (1970s and 1980s), derivatives on Eurodollars took over thereafter, and became the dominant market with the greater trading volume. Moreover, IRDs on Eurodollars are often simpler than IRDs on Treasuries. Recall that Eurodollars are dollar deposits in European banks (or European subsidiaries of US banks) that earn interest in dollars. They were concocted during the 1950s to evade the jurisdiction (and thus regulatory controls) of the US government. Moreover, unlike Treasury securities, for which large auctions affect the available supply, the larger supply of dollars changes less frequently and is less “lumpy.” Less regulation and a less managed supply enable Eurodollar rates to adjust more freely to market forces than do Treasuries. This makes Eurodollar markets attractive to hedgers and speculators. These factors have contributed to the tremendous growth of Eurodollar markets, which are measured in trillions of dollars. Recall from Chapter 2 that there are three major interest rates relevant to Eurodollars. First is the Eurodollar rate, which is the interest rate paid on Eurodollar deposits. Second is the generic London Interbank Offer Rate, which is popularly known as libor. It is the interest rate at which major London banks offer to lend funds (in different currencies) to each other. Libor quotes on Eurodollars are among the most popular rates in the interest rate markets. Third is an international interest rate index based on average libor rates. The International Continental Exchange (ICE) gets quotes from numerous banks as to their offer rates on Eurodollar deposits of all maturities ranging from one day to one year. For each maturity deposit, some of the highest and lowest quotes are dropped, and the rest are averaged to compute a libor rate index. Such indexes are computed for many currencies, with multiple maturities for each. This libor rate index for Eurodollars is the most widely used benchmark or reference rate for short-term interest rates worldwide. As it has some credit risk,

499

500

CHAPTER 21: YIELDS AND FORWARD RATES

this libor index rate exceeds the corresponding rate on a similar maturity Treasury security, the difference being known as the Treasury–Eurodollar (TED) spread. Because an index is harder for any single financial institution to manipulate, this libor rate index is used as the reference rate for over-the-counter interest rate derivatives contracts.2 Like Treasury securities, Eurodollar interest rate computations assume 360 days in a year.

Yields and the Yield Curve

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In Chapter 2, we introduced zero-coupon bonds for the primary purpose of moving cash across time. Now we employ zeros for extracting the necessary interest rates. We only consider zero-coupon bonds that are default-free. Extensions of these insights to risky bonds are discussed briefly in Chapter 26. For simplicity, we denote time as progressing in unit increments t = 0, 1, 2, 3, . . . Of course, in practice, one needs to replace the unit increment with the actual days between the relevant cash flows. From now on, we also need to attach time notations to our familiar symbols. For example, the price at time t of a zero-coupon bond that pays a dollar at time T is denoted B(t, T). A zero-coupon bond’s price at time 0 is denoted B(0, T). This notation is illustrated in Figure 21.1. The yield (or yield to maturity or internal rate of return) on a zero-coupon bond is that interest rate that equates the discounted bond’s cash flows to its current price. It is the interest rate earned each period if the zero-coupon bond is held until maturity. The yield of a default-free zero is sometimes called the spot rate, although this terminology is fading in usage.

FIGURE 21.1: A Zero-Coupon Bond’s Cash Flow and Prices

Time

0

B(0, T)

1

2

…

t B(t, T)

…

T $1

B(t, T) is time t price of a zero-coupon bond that pays a dollar at time T.

2

However, the libor index was nonetheless manipulated during the financial crisis of 2007.

YIELDS

In symbols, the yield R(0, T) at time 0 (today) is defined by the equation B (0, T) =

1 [1 + R(0, T)]T

(21.1)

Notice that yields generalize the concept of a dollar return used before. Example 21.1 shows how to compute yields and create a yield curve that plots the zero-coupon bonds’ yields (on the vertical axis) as a function of maturity (on the horizontal axis). This example also introduces the Common Zero-Coupon Bond Price Data utilized in this and subsequent chapters.

EXAMPLE 21.1: Computing the Yield Curve ■

Today is time 0. Consider zero-coupon bond prices with different maturity dates time T denoted by B(0, T), where T = 1, 2, 3, 4. The common zero-coupon bond price data are shown in Table 21.1. - The first column reports the various times to maturity T = 1, 2, 3, 4. The longest-term zerocoupon bond matures at time 4. - The second and third columns give today’s zero-coupon bond prices: B(0, 1) = $0.97, B(0, 2) = $0.93, and so on.

■

These numbers are used to compute the yields in the fourth and fifth columns. A rearrangement of Equation 21.1 gives the yield as

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R (0, T) =

1 − 1. [B(0, T)]1/T

- For example, the yield on the two-period zero is R(0, 2) = 1/[B(0,2)]1/2 – 1 = 0.0370. - A plot of these yields against maturities gives the yield curve. In this case, it is also called the zero-coupon yield curve (or zero curve) (see Figure 21.2).

TABLE 21.1: The Common Zero-Coupon Bond Price Data and Yields Times to Maturity T

Today’s Zero-Coupon Bond Prices, B(0,T)

Yields, R(0,T)

1

B(0, 1)

0.97

R(0, 1)

0.0309

2

B(0, 2)

0.93

R(0, 2)

0.0370

3

B(0, 3)

0.88

R(0, 3)

0.0435

4

B(0, 4)

0.82

R(0, 4)

0.0509

501

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CHAPTER 21: YIELDS AND FORWARD RATES

FIGURE 21.2: Zero-Coupon Yield Curve Yields 0.06 0.05 0.04 0.03 0.02 0.01 0.00 1

2

3

4 Time to maturity

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Several concepts are related to yields. ■

Standing at time t, the yield on the shortest-maturity zero-coupon bond maturing at time (t + 1) has a special name: the spot rate of interest (or the spot rate). Using our notation for the yield, this is R(t, t + 1), but for simplicity, we write it as R(t). We emphasize that whenever the second argument of the yield notation is missing, it will always denote the spot rate of interest.

■

For future reference, we note that a riskless money market account earns the spot rate of interest. Standing at time t, a money market account’s return is known over the next time step [t, t + 1]. Of course, we do not know at time t what the time (t + 1)’s or any future time period’s spot rate will be. Future spot rates of interest are random.

■

The graph of the yield curve is called the term structure of interest rates. Figure 21.3 graphs the yield curve at five different dates for Treasury securities. Note that the yield curve can be flat or upward or downward sloping. The shape of the yield curve reflects various factors such as (1) expectations of the market participants regarding the future movement of interest rates, (2) risk aversion of traders, and (3) the supply and demand considerations of the underlying zero-coupon bonds. The hypothesis that the shape of the yield curves only reflects expectations regarding the future movements of interest rates is known as the expectations hypothesis. Extension 21.2 provides a brief discussion of the expectations hypothesis as well as several other competing theories explaining the term structure of interest rates.

YIELDS

FIGURE 21.3: Yield Curves for Treasury Securities at Selected Dates Yields 18% 16% 14% 12%

1981-May 1981-Aug 1987-Nov 1995-Nov 1998-Sept

10% 8% 6% 4% 2% 0.08333 0.25 0.5 1 2 3 4 5 10 15 20 25 30

0%

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Time to maturity

Consider a default-free coupon bond that trades in the market at time 0. The yield (or yield to maturity, also called the internal rate of return) on the coupon bond is the interest rate that equates the discounted coupon bond’s cash flows to the bond’s current price PT . In symbols, the yield y satisfies PT =

C C C C+L + +…+ + 2 T−1 (1 + y) (1 + y)T (1 + y) (1 + y) T

=

C L + ∑ (1 + y)t (1 + y)T t =1

(21.2)

where C denotes the yearly coupon, L denotes the principal (or par or face value), and T denotes the time to maturity in years.3 One can interpret the yield as the interest rate earned per period if (1) one invests today in a coupon bond that matures at time T, (2) one holds the bond until maturity, and (3) all the coupons received get reinvested (in the future) at today’s yield y. Of course, of these three conditions, the third is the most unreasonable because future interest rates are random, and they will be unlikely to equal today’s yield.

3

We assume annual coupon payments for simplicity. When coupons are paid every six months, the formula is modified with C/2 replacing C, the equation in the denominator becoming (1 + y/2), and the annual yield is written as (1 + y/2)2 – 1.

503

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CHAPTER 21: YIELDS AND FORWARD RATES

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EXTENSION 21.2: The Expectations Hypothesis The classical (macro) economics term structure of interest rate literature characterizes the risk premium embedded in the term structure of interest rates according to hypotheses about investor behavior: the market segmentation hypothesis, the liquidity preference hypothesis, and the expectations hypothesis. Recall that risk premia are the compensation (in terms of an increase in expected returns above the spot rate) paid for the risks embedded in different securities. Given an understanding of risk premia, one can both price zero-coupon bonds and forecast future spot rates of interest. The market segmentation hypothesis, which is normally associated with Professor John M. Culbertson (1957), states that the different maturity equilibrium zero-coupon bond prices are determined in isolation of each other by distinct market clienteles demanding payoffs at different horizons. Thus there is market segmentation across different future time horizons. This implies that the risk premia paid for bonds of different maturities are determined in isolation of the risk premia paid for the other maturity bonds, and that there might be arbitrage across market segments. In contrast, Nobel laureate economist Sir John Richard Hicks’s (1946) liquidity preference hypothesis is that the equilibrium zero-coupon bond prices of different maturities are jointly determined by supply and demand considerations, in particular, liquidity needs across time, and the interest rate risk premia reflect these considerations. Professors Franco Modigliani and Richard Sutch’s (1966) preferred habitat theory argues that borrowers and lenders have their own preferred maturity habitats (as suggested by market segmentations hypothesis) and would only move to another habitat if they were given an increased premium to compensate for the risk and cost of moving. Modern equilibrium models of the term structure of interest rates can be viewed as the representation of the liquidity preference (or preferred habitat) hypothesis in a continuous time framework (see Cox et al. 1985; Dunn and Singleton 1986; Sundaresan 1984). The expectations hypothesis is associated with economists Irving Fisher (1930) and Friedrich A. Lutz (1940). It has as its basic premise that zero-coupon bonds of different maturities are perfect substitutes. In the literature, this perfect substitutes hypothesis has been formalized (in terms of the mathematics) in three distinct ways. Each of these formalizations can be characterized by a formula for the zero-coupon bond’s price. Jarrow (1981) and Cox et al. (1981) independently discovered this critique of the various expectations hypotheses. The following discussion is presented for the general case in terms of today (time 0), an intermediate date (time t), and the maturity date for a long-term bond (time T). The local expectations (LE) hypothesis is that Et

[

B (t + 1, T) − B (t, T) = R (t) ] B (t, T)

(1a)

where Et [.] denotes expectation, B(t, T) is the time t price of a zero-coupon bond that pays a dollar at maturity date T, and R(t) is the spot rate at time t (which is the one-period rate for immediate borrowing or lending; see Equation 21.1). This formalization interprets “perfect substitutes” as meaning that the different maturity zero-coupon bonds earn no risk premia, that is, the expected return on all zero-coupon bonds is equal to the spot rate of interest.

YIELDS

505

Equivalently, in terms of bond prices, the LE hypothesis can be written as: B (0, T) = E

1 { [1 + R (0)] [1 + R (1)] … [1 + R (T − 1)] }

(1b)

This implies, as in Equation 1a, that the T-maturity bond’s price is its expected discounted payoff using the spot rate of interest with no adjustment for risk. The return-to-maturity expectations (RME) hypothesis is that 1 − 1 = E {[1 + R (0)] [1 + R (1)] … [1 + R (T − 1)]} − 1. B (0, T)

(2a)

This formalization of the “perfect substitutes” condition is that the T-period holding period return from a zerocoupon bond (the left side of Equation 2a) is equivalent to the expected return from holding the money market account over the same horizon (the right side of Equation 2a). To understand this second assertion, looking at the right side of Equation 2a, the first term corresponds to the expected interest earned by investing a dollar in a money market account at time 0 and letting it roll over until time T. Interest is earned on interest. Subtracting the original dollar gives us the expected percentage return on the money market account over [0, T]. Equivalently, in terms of bond prices, the RME hypothesis can be written as

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B (0, T) =

1 E {[1 + R (0)] [1 + R (1)] … [1 + R (T − 1)]}

(2b)

It can be shown that the LE and RME hypotheses are mutually exclusive (unless interest rates are deterministic) by Jensen’s inequality. The unbiased expectations (UE) hypothesis is that f (0, t) = E {R (t)} .

(3a)

where f (0, t) is a one-period borrowing or lending rate that you can contract at time 0 for the time interval [t, t + 1]; see Equation 21.9 later in the text. Here the perfect substitutes condition is formalized to mean that forward rates provide an unbiased estimate of the expected future spot rate. Equivalently, in terms of bond prices, the UE hypothesis can be written as B (0, T) =

1 . [1 + R (0)] [1 + E {R (1)}] … [E {R (T − 1)}]

(3b)

It can also be shown that the LE and UE hypotheses are mutually exclusive. If the term structure of interest rates has risk premia, then all three forms of the expectations hypotheses are not true, and the pricing of zero-coupon bonds becomes more complex. Unfortunately, empirical evidence strongly supports the existence of interest rate risk premium and the rejection of all three forms of the expectations hypothesis. But they are interesting hypotheses none-the-less because they help us to better understand the pricing of zero-coupon bonds. Knowing what is not true sometimes helps us understand what is, and why!

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CHAPTER 21: YIELDS AND FORWARD RATES

EXAMPLE 21.2: Computing a Coupon Bond’s Yield ■

Consider a two-year coupon bond with price P2 = $101.86, principal L = $100, and a yearly coupon of C = $6. Using Equation 21.2, the coupon bond’s yield y is the solution to the equation 6 106 , + 2 (1 + y) (1 + y) or, 101.86y2 + 197.72y − 10.14 = 0.

101.86 =

■

Solving this quadratic equation (analytically, by trial and error, or by numerical methods) gives4

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y = 0.05 or 5 percent. ■

Microsoft Excel enables you to compute the yield with one formula, which has the syntax: YIELD(settlement,maturity,rate,pr,redemption,frequency,basis).

■

The inputs are as follows: - Settlement is the bond’s settlement date—the day from which the bond earns interest. Let us assume it is January 15, 2012, which is input into Excel as DATE (2012,1,15). - Maturity is the date the bond expires, or January 15, 2014. - Rate is the annual coupon rate, or 6 percent. - Pr is the bond’s price per $100 face value, which is $101.86. - Redemption is the bond’s redemption value per $100 face value, which is $100. - Frequency is the number of coupon payments per year, 1 for annual payments, 2 for semiannual payments, 4 for quarterly payments, and so on. - Basis is the type of day count basis to use. Here we use 0 to denote that we are assuming 30 days in a month and 360 days in a year. Other examples of basis are actual/actual, actual/360, and so on. The yield is given by YIELD(DATE(2012,1,15), DATE(2014,1,15),6%,101.86,100,1,0) = 0.049999687

4

A quadratic equation ax2 + bx + c = 0 has the solutions x = [−b ± √(b2 − 4ac)]/2a. Select the positive value for y.

A special case when computing a yield should be noted. If the price of the bond in Example 21.2 were 100 instead (equal to par), then the yield would equal the coupon rate of 0.06. When this happens, the coupon rate is the same as the yield and is called the par bond yield. Yields are often used to select bonds for inclusion into an investment portfolio. However, this approach to bond portfolio selection is problematic. Just like warnings on cigarette cartons that alert one to the dangers of smoking, a warning needs to be posted at this point with respect to using yields for bond portfolio management. As noted earlier, the yield on a coupon bond is equivalent to the internal rate of return (IRR). The IRR is a well-studied object in capital budgeting, which is the science of choosing among investment alternatives (in our case, bonds). The recommendation

THE TRADITIONAL APPROACH

in early texts was to select projects with the highest IRR (read as yields). However, in their classic Principles of Corporate Finance, Brealey and Myers (2003) debunk IRR’s desirability and point out several pitfalls. The most critical pitfall from our point of view is that when using IRR in making investment decisions, one is assuming that the reinvestment rate for future cash flows is non-random and equal to the IRR. Of course, in actual markets, interest rates are random, and this assumption is violated. Brealey and Myers therefore recommend not using the IRR when making investment decisions, and they provide suitable alternatives. As a consequence, astute readers like you are also warned against using a bond’s yield (the IRR) as the sole determinant for choosing among different bonds to hold in an investment portfolio. We will provide suitable alternative strategies for bond selection in Chapters 23–25, in the context of the Heath–Jarrow–Morton (HJM) framework.

21.3

The Traditional Approach

In 1938, Frederick R. Macaulay proposed the concept of duration (also known as Macaulay’s duration), which led to the development of the traditional approach to interest rate risk management. We next discuss how duration and its related concept, convexity, are used for fixed-income securities portfolio management.

Duration First, using Equation 21.2, we write the bond’s price as Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

T

P=

L C + T ∑ 1+y t t =1 ( ) (1 + y)

(21.3a)

where y is the bond’s yield. Now assume that there is a small change in the bond’s yield, Δy, which equals the new yield minus the original yield. Next, use a Taylor series expansion to get an equation for the approximate change in value of the bond’s price:5 1 C L + Δy ∑T 1 + y [ t =1 (1 + y)t (1 + y)T ] Δy = −PDur 1+y

ΔP ≈ −

5

(21.3b)

A Taylor series expansion allows us to write the change in the bond price as P (y + Δy) − P (y) = (dP/dy) (Δy) + (1/2) (d2 P/dy2 ) (Δy)2 + …

(21.1)

This relation is an approximation because only the first term on the right side is retained. We get a better approximation if the second derivative is used. This takes us to the notion of “convexity” (which we introduce later), but it also increases computational complexity.

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CHAPTER 21: YIELDS AND FORWARD RATES

where ≈ denotes approximation and Dur is duration, where T

Dur =

1 L C ×t+ ×T t ∑ ] [ [ P { t =1 (1 + y) (1 + y)T ] }

(21.4)

A bond’s duration (Equation 21.4) has a useful economic interpretation. The terms in the braces in Equation 21.4 are the discounted cash flows. They become “weights” when expressed as a fraction of the bond price. These weights add up to 1, giving Equation 21.4 the interpretation of a weighted average. In the summation, the weights are averaging the times to the cash payments t = 1, 2, … T. Thus duration can be interpreted as the bond’s average life. Using this interpretation, it is easy to see that the duration of a zero-coupon bond is the same as the time to maturity because there are no early cash flows and the equation within the braces is just the zero-coupon bond’s price times the time to maturity. Equation 21.3b is rewritten as the following result (see Result 21.1).

RESULT 21.1 A Modified Duration Formula for a Bond’s Price Change

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For a small change in the bond’s yield to maturity y, the change in the bond’s price is ΔP ≈ −PDurM Δy (21.5) where P is the original bond price, ΔP is the change in the bond price (Δ denotes New value – Original value), DurM ≡ Dur/(1 + y) is the modified duration, and Δy is the change in the yield to maturity.

Equation 21.4 shows that when yields change, the change in the bond’s price can be approximated as minus the change in the bond’s yield times the product of the bond’s price and its modified duration. The negative sign in Equation 21.5 reflects the fact that prices and yields move in opposite directions. The formula also shows that all else constant, a bond with a higher modified duration will have a greater price change for a given change in the yield. As such, modified duration can be interpreted as a “risk measure” for the bond. The next example shows how to compute duration, modified duration, and to approximate changes in the bond’s price when yields change only slightly.

THE TRADITIONAL APPROACH

EXAMPLE 21.3: Using Modified Duration to Approximate a Bond’s Price Change ■

Let a newly issued two-year coupon bond have a par value of $100, a coupon rate of 6 percent ($6), and a yield y = 0.05 or 5 percent per year. We present the following computations in Table 21.2: - Columns 1 and 2 record the timing and magnitude of the cash flows. - Column 3 reports the present value of the cash flows. They are computed by dividing $6 by (1 + y) = 1.05 and $106 by (1 + y)2 , respectively. Sum these numbers to get the bond’s price, $101.8594. This is the same as the bond price P2 in Example 21.2. - Column 4 is presented for computational convenience. The entries in each cell are derived by multiplying the present value of the cash flow by its timing (column 3 times column 1). Their sum equals $198.0045. Dividing this number by the bond price gives the duration: Dur =

■

198.0045 = 1.9439 years 101.8594

Suppose that the yield increases by 25 basis points. Discount the bond’s cash flow by the new rate 5.25 percent to get the new bond price of $101.3896. The bond price change equals ΔP = New price − Old price = −$0.4698

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■

Using Equation 21.5 instead, we compute the bond’s price change as ΔP ≈ −P DurM Δy = −101.8594 × = −$0.4714.

1.9439 × 0.0025 1.05

Comparing the two changes, the bond price actually declines to $0.4698 but the duration-based measure approximates a decline of $0.4714. The error is a little over 0.15 cents on a principal amount of $100.

TABLE 21.2: Bond Price and Duration Computation Time t (years)

Cash Flows

Present Value of Cash Flows

Present Value of Cash Flow × t

1

6

5.7143

5.7143

2

106

96.1451

192.2902

$101.8594 (= bond price)

198.0045

Sum

Duration, Dur = 198.0045/101.8594 = 1.9439 years.

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CHAPTER 21: YIELDS AND FORWARD RATES

Notice an analogy. Just as we had used delta to get a linear approximation to changes in options prices for small changes in stock prices, we use modified duration to get a linear approximation to changes in bond prices for small changes in the yield. Both of these approaches are based on a first-order Taylor series expansion, and just as we improved the approximation in the case of options by incorporating the second derivative gamma, we can do the same and get a better approximation to the price-yield curve by introducing the second derivative with respect to the bond’s yield. This gives a price change formula that uses both duration and convexity. Because the convexity approximation is identical to that obtained when using an option’s gamma, we omit the details and refer the reader to the relevant discussion in Chapter 20.6

Modified Duration Hedging Consider a bond portfolio consisting of ni units of the coupon bond with coupon payment Ci , maturity Ti , principal Li , and price Pi for bonds i = 1, . . . N. The time 0 value of this portfolio is N

V=

∑

ni Pi .

(21.6a)

i =1

The bond portfolio’s modified duration is7 N

DurMV =

∑

wi DurMi

(21.6b)

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i =1

where wi =

ni Pi and DurMi is the modified duration of bond i. Note that the weights V N

sum to 1, that is

∑

wi = 1.

i =1

The bond portfolio’s modified duration is just the weighted average of the individual bond’s modified durations, where the weight for bond i corresponds to the percentage of the portfolio’s value represented by the position in that bond. Given an existing bond portfolio, modified duration hedging is the process of modifying the composition of the bond portfolio through buying, selling, and/or shorting bonds such that the new portfolio has a zero-modified duration. Under certain conditions (to be shown below), duration hedging will remove the interest rate risk from a bond portfolio. 6

The new price change formula is ΔP ≈ −PDurM Δy + (P/2) Convexity(Δy)2 , where we define T

Convexity =

t(t + 1)C T(T + 1)L 1 + . t ] [ (1 + y) P(1 + y)2 {∑ (1 + y)T } t =1

7 One can derive Equation 21.6b as follows. Define y to be the yield on portfolio V, viewing V as a “single bond” whose coupons differ across time. Take the derivative of both sides of Equation 21.6a with respect to y. This uses Equation 21.3, which gives P as a function of an arbitrary yield y. Next, use Equation 21.5 for each bond, including V when viewed as a “bond” by itself, and algebra to obtain Equation 21.6b. In Equation 21.6b, each bond’s modified duration is evaluated at the portfolio’s yield y, and not their own yields.

THE TRADITIONAL APPROACH

During the last few decades of the 20th century, duration hedging was a widely used technique for managing the interest rate risk of a bond portfolio. However, it is currently being supplanted by the more sophisticated hedging techniques based on the HJM model. To understand the conditions under which traditional duration hedging removes the interest rate risk from a bond portfolio, let’s consider the simplest bond portfolio consisting of: (1) a single coupon-bond with coupon payment C, maturity T, principal L, and price P and (2) n zero-coupon bonds with maturity T and price B. We note that the coupon bond has a modified duration of DurM and that the T-period zero-coupon bond has a modified duration of T/(1 + yB ), where yB is the yield on the zero-coupon bond. The portfolio’s value is V = P + nB. (21.7a) To duration hedge we choose n such that the right side of Equation 21.7b is zero, which implies that the modified duration of the portfolio is approximately zero,8 that is nB T P = 0. (21.7b) DurMV ≈ ( ) DurM + ( ) V V 1 + yB Algebra gives the number of zero-coupon bonds to trade as:

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n=−

P × DurM × (1 + yB ) . B×T

(21.7c)

This is called the modified-duration hedge ratio. Now let’s study the conditions under which this zero-modified duration portfolio’s value does not change when interest rates change. First, we can write the change in the portfolio’s value as ΔV = ΔP + nΔB. (21.8a) Second, using the modified-duration approximation given in Result 21.1 for both the coupon and zero-coupon bonds, we can rewrite this as ΔV ≈ −PDurM ΔyP − nB

T Δy 1 + yB B

(21.8b)

where ∆yP is the change in the yield on the coupon-bond P and ∆yB is the change in the yield on the zero-coupon bond B. Last, substitution of the modified-duration hedge ratio n from Equation 21.7c yields: ΔV ≈ −PDurM (ΔyP − ΔyB ) .

8

(21.8c)

The duration of the portfolio V is only approximately equal to zero because we replace the portfolio’s yield y on the right side of Equation 21.7b with both the bond’s and zero-coupon bond’s yields, respectively, when computing their durations. The hedge ratio in Equation 21.7c is the appropriate one to remove interest rate risk, as shown below.

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CHAPTER 21: YIELDS AND FORWARD RATES

This is the result for which we were looking! Examining this equation, we see that the zero-modified-duration bond portfolio’s value will be neutral to changes in interest rates if and only if the coupon- and zero-coupon bonds’ yields are equal when interest rates change, that is ∆yP = ∆yB . This is a very strong condition. It implies that modified-duration hedging is effective if and only if the yield curve shifts in a parallel fashion. At this point, you can rightly sound an alarm. If you recall Figure 21.3, you see that the yield curve does not evolve through time via parallel shifts! Consequently, modified-duration hedging doesn’t really work, especially when the long and short ends of the term structure of interest rates behave differently. But modified-duration hedging has another problem as well. Recall that modifiedduration hedging only uses the first-order approximation from the Taylor series expansion of the bond’s price as a function of the yield. The first order approximation only works well when yield changes are small. If yield changes are large, then the second-order approximation in the Taylor series expansion needs to be included. This is called convexity hedging. Because convexity hedging with bonds is analogous to gamma option with options, we omit the remaining details and refer the reader to the relevant discussion in Chapter 20. Unfortunately, convexity hedging does not solve the non-parallel shifting yield curve problem. For this reason, we do not recommend its use.

Applications and Limitations

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Duration, its modifications and extensions, has been widely used in the past to manage interest rate risk. 1. Active bond portfolio management. One can modify a portfolio’s duration to bet on the direction of future interest rate movements. For example, if one believes that interest rates are going to fall (which would increase the value of a bond portfolio because bond prices will rise), just increase the portfolio’s duration by eliminating low duration bonds and buying high duration bonds. High duration bonds are more sensitive to interest rate changes, so that an increase in the portfolio’s duration will increase profits as interest rates fall. Do the opposite to soften the blow of an interest rate hike. If desired, one can construct the portfolio to be duration neutral, that is, to have zero duration. As noted previously, making a bond portfolio duration neutral reduces, but does not eliminate, interest rate risk from the bond portfolio. 2. Matching assets and liabilities. A typical bank has liabilities (like checking and savings accounts) with very low duration (because customers can remove them at a moment’s notice) and assets (like loans to consumers and businesses) with high durations (because they are typically for longer periods). In its natural state, a bank holds a speculative position in terms of the shape of the yield curve. A bank can

FORWARD RATES

reduce this vulnerability by managing the gap between the assets’ and liabilities’ durations. For example, it can increase the duration of the liabilities by encouraging customers to invest in long-term deposits, and it can lower the duration of the assets by issuing floating-rate loans such as adjustable-rate mortgages and lines of credit that charge variable rates tied to short-term interest rates. Duration-based hedging has many shortcomings that limit its efficacy. Duration hedging strategies do not completely eliminate interest rate risk from a bond portfolio because of the inherent weaknesses of this approach. First, it only works for small changes in interest rates, and in reality, interest rates sometimes change by larger magnitudes before one can rebalance the portfolio. Second, it only works for parallel shifts of the yield curve, which assumes that all of the bonds’ yields change by the same amount. However, this rarely happens in practice because short-term interest rates are much more volatile than long-term interest rates (see Figure 21.3). Combined, these two weaknesses greatly reduce the effectiveness of duration hedging. Although some traders may still use the concept of duration to manage a bond portfolio’s interest rate risk, duration-based hedging is on its way out. Term structure modeling, which we develop in this and the subsequent chapters, is a far superior method for interest rate risk management because it works for all possible evolutions of the yield curve.

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21.4

Forward Rates

“The old order changeth, yielding place to new,” reflected a dying King Arthur [in Alfred Lord Tennyson’s poem Le Morte D’Arthur (1845)]. So is it with respect to duration and traditional interest rate risk management. Newer techniques have been developed that overcome all of duration hedging’s limitations. These techniques consider the impact of the changing shapes of the term structure of interest rates on bond prices. The approach is based on the HJM model discussed in Chapters 23–25. This section introduces the concept of a forward rate, which forms the basis of modern interest rate risk management. Forward rates are handy for several reasons. 1. Forward rates are unit-free. Rates provide a better measure to compare across bonds than do bond prices. To see this, consider a dollar increase in two identical stocks where one had a 2:1 split the day before. Does a dollar stock price increase convey the same meaning after the split as it did before? The answer is no because the benefit of a $1 increase in the stock’s price depends on its initial price. Yet a 1 percent change for both stocks either before or after the split is comparable because it is unit-free. The same is true for forward rates when considering changes in bond prices.

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2. Forward rates are more stable than bond prices. Again, that rates are unit-free becomes relevant. Consider a stock that earns 10 percent return per year, a stable rate. Its prices are $100 today, $110 after one year, $121 after two years, $133.10 after three years, and so on. The price changes differ across time and are not stable; however, the 10 percent return is constant. Similarly, zero-coupon bond prices change across time, yet forward rates can be stable. These considerations become important in estimating and modeling the term structure of interest rates in Chapters 23–25. 3. Versatility of forward rates. The building blocks of modern interest rate management theory are interlinked: Zero − coupon bond prices ↔ Yields ↔ Forward rates where we use ↔ to indicate that given one of these quantities, the others can be computed. Since all of these quantities are interchangeable, we select the simplest for analysis. This takes us to forward rates.

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The Definition To understand the definition, let us first study an example. Consider two zero-coupon bonds trading today (time 0) with maturities at times 3 and 4, and with prices B(0, 3) = $0.88 and B(0, 4) = $0.82, respectively. Their price ratio adjusted by subtracting 1 is B(0, 3)/B(0, 4) – 1 = 0.0732, or 7.32 percent. This adjusted ratio captures the implicit interest earned on the four period bond between times 3 and 4. Indeed, the extra interest earned over the time period [3, 4] is the only difference between the three and four period zero-coupon bonds. Extending this example, we can formally define the time 0 forward rate for the future time period [T – 1, T] as f (0, T − 1) =

B (0, T − 1) −1 B (0, T)

(21.9)

where B(0, T– 1) and B(0, T) are today’s zero-coupon bond prices for bonds maturing at times (T –1) and T, respectively.9 As in the example, one can understand the forward rate by looking at Figure 21.4. This figure shows that the forward rate f (0, T – 1) isolates the implicit interest earned on the longer maturity bond over the last time period [T – 1, T]. Example 21.4 illustrates this computation.

9 We could have written the symbol as f (t,T – 1,T) to denote the forward rate, but we do not need the last symbol T because this is a one-period rate.

FORWARD RATES

FIGURE 21.4: The Forward Rate

f(0, T – 1) Time 0

1

2

…

t

…

(T – 1)

T

$1

B(0, T – 1)

$1

B(0, T)

Forward rate f(0, T – 1) is the one-period rate that you can contract at time 0 for investing a dollar over the time horizon [T – 1, T]

EXAMPLE 21.4: Computing Forward Rates

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Table 21.3 uses the data from Example 21.1. ■

The first column reports the various zero-coupon bond maturities denoted by T = 0, 1, 2, 3, 4. The second and third columns give today’s zero-coupon bond prices. B(0,0) is trivially 1.

■

These numbers are used to compute the forward rates, reported in the fourth and fifth columns. By Equation 21.9, f (0,0) = B(0,0)/B(0,1) – 1 = 0.0309. Similarly, one can compute the other rates such as f (0,2) = [B(0,2)/B(0,3)] – 1 = 0.93/0.88 – 1 = 0.0568, and so on.

TABLE 21.3: Zeros, Forward Rates, and Yields Maturity

Today’s Zero-Coup on Bond Prices

Today’s Forward Rates

Yields, R(0, T)

0

B(0, 0)

$1

f (0, 0)

0.0309

R(0, 1)

0.0309

1

B(0, 1)

0.97

f (0, 1)

0.0430

R(0, 2)

0.0370

2

B(0, 2)

0.93

f (0, 2)

0.0568

R(0, 3)

0.0435

3

B(0, 3)

0.88

f (0, 3)

0.0732

R(0, 4)

0.0509

4

B(0, 4)

0.82

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CHAPTER 21: YIELDS AND FORWARD RATES

■

Figure 21.5 plots the forward rate curve that depicts the forward rates versus time to maturity. For comparison, also shown is the zero-coupon bond yield curve (from Example 21.1).

■

We will often use the zero-coupon bond prices, forward rates, and yields to maturity from this table in subsequent numerical examples.

FIGURE 21.5: Forward and Zero-Coupon Yield Curves Forward Rates and Yields 0.08 0.07 0.06 0.05 Forward Rate Yield

0.04 0.03 0.02 0.01 0 1

2

3

4

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Time period

Understanding Forward Rates We can understand forward rates in three related ways: 1. The first interpretation follows from the definition, which is the conventional approach in the academic literature and finance industry. Taking the ratio of two consecutive zero-coupon bond prices gives the implicit interest rate earned over the relevant future time period. 2. A second interpretation is as the rate one can contract today on riskless borrowing or lending over the future time period [T – 1, T]. One can see this interpretation using a payoff table (see Table 21.4), where the numbers from Example 21.4 are employed to illustrate how to replicate the borrowing rate over a future time period by trading in zero-coupon bonds today. The procedure is as follows: a. Today is time 0 and the longest bond’s maturity date is T = 4. b. Today short one unit of the three-period zero-coupon bond worth B(0,3) = $0.88 and buy [B(0,3)/B(0,4)] = 0.88/0.82 = 1.0732 units of the four-period zero-coupon bond worth B(0,4) = $0.82. c. This portfolio has zero value today and hence zero cash flow. d. The first cash flow occurs at time 3 and is a negative dollar. This is the same as borrowing $1 at time 3.

FORWARD RATES

e. The next cash flow is at time 4, when the four-period zero-coupon bond matures. This is a positive cash flow of [B(0,3)/B(0,4)] = $1.0732, which equals [1 + f (0,3)] by the definition of a forward. This is equivalent to paying back the borrowing of $1 plus interest. To obtain the general result, replace time 4 by time T. This substitution shows that the cash flows obtained from the zero-coupon bond portfolio in the payoff table exactly replicate the cash flows from contracting at time 0 to risklessly borrow a dollar over [T – 1, T]. 3. The third interpretation is presented in the next section, where we show that the forward rate corresponds to the rate quoted on a forward rate agreement contract.

TABLE 21.4: Payoff Table for a Portfolio of Zero-Coupon Bonds That Generates the Forward Rate Portfolio

Time 0

Time T – 1 = 3

Short one unit of a three-period zero-coupon bond worth B(0,3) today B(0, 3) 0.88 = = Buy B(0, 4) 0.82 1.0732 units of a four-period zero-coupon bond worth B(0,4) today

B(0, 3)

–1

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Sum

−

Time T= 4

B(0, 3) B(0, 4) B(0, 4)

0

B(0, 3) B(0, 4)

–1

1.0732 =

B(0, 3) = f(0, 3) + 1 B(0, 4)

Using Forward Rates This section shows how you can express other rates and prices in terms of forward rates. The next chapter will show you how to do the same with swaps, and Chapters 23–25 will reveal the critical role forward rates play in modern interest rate derivatives pricing models. We start with an example (see Example 21.5).

EXAMPLE 21.5: Computing a Zero-Coupon Bond Price Using Forward Rates ■

Continuing with our previous example, using definition (21.9), we can write the first zero-coupon bond’s price in terms of forward rates: 1 − 1, B (0, 1) 1 or B (0, 1) = 1 + f (0, 0) f (0, 0) =

(21.10a)

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CHAPTER 21: YIELDS AND FORWARD RATES

where f (0, 0) is also the spot rate of interest. ■

Considering a forward rate that starts one period later, at time 1, we get B (0, 1) − 1, B (0, 2) 1 1 1 or B (0, 2) = B (0, 1) = 1 + f (0, 1) [1 + f (0, 0)] [1 + f (0, 1)] f (0, 1) =

(21.10b)

where the last term on the right uses the value of B(0,1) from Equation 21.10a. ■

We can generalize by continuing these successive substitutions to derive an equation for a longermaturity zero-coupon bond.

The generalization of this example is our next result (see Result 21.2).

RESULT 21.2 A Zero-Coupon Bond’s Price and Yield in Terms of Forward Rates

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B (0, T) =

1 [1 + f (0, 0)] [1 + f (0, 1) … [1 + f (0, T − 1)]

R (0, T) = ([1 + f (0, 0)] [1 + f (0, 1)]

1

… [1 + f (0, T − 1)]) T − 1

(21.11a)

(21.11b)

where B(0,T) is the time 0 price of a zero-coupon bond maturing at time T, R(0,T) is the zero-coupon bond’s yield to maturity, and f (0, t) represents the forward rate contracted at time 0 for [t, t + 1].

Equation 21.11a expresses a zero-coupon bond’s price B(0, T) as the discounted value of a dollar received at time T, where the discounting is done by the relevant forward rates. Equation 21.11b states that a zero coupon bond’s yield is the geometric average of the corresponding forward rates minus 1. This result follows readily from the definition of a zero coupon bond’s yield as given in Equation 21.1 at the start of this chapter. Equation 21.11b also provides an additional insight: the spot rate of interest (denoted by R[0]) is the forward rate for immediate borrowing and lending for one period, that is, R (0) = R (0, 1) = f (0, 0) .

(21.12)

FORWARD RATES

519

Although zero-coupon bonds are easier to work with, they don’t often trade. In this case one can determine forward rates from coupon bond prices. This method is demonstrated in Extension 21.3. The concepts of forward rates and yields are useful for understanding the basic interest rate derivatives, forward rate agreements, and interest rate futures, which are discussed in the next section.

EXTENSION 21.3: Computing Forward Rates from Coupon Bond Prices Computing the forward rates from coupon bond prices is a simple exercise in solving a system of linear equations for a collection of unknowns. First, we need the following no-arbitrage relation for a coupon bond. Consider a coupon bond with price PT , principal L, maturity date T, and a coupon payment C paid every period. To avoid arbitrage, in a frictionless and competitive market, it must be true that

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PT = CB (0, 1) + CB (0, 2) + … + CB (0, T − 1) + (C + L) B (0, T) .

(1)

This follows by noting that the cash flows from a portfolio of zeros consisting of C units of B(0,1), C units of B(0,2), . . . , C units of B(0, T– 1), and (C + L) units of B(0, T) are the same as from the coupon bond. Hence the cost of constructing this portfolio must equal the price of the traded coupon bond, or else there is an arbitrage opportunity. The right side of Equation 1 is the cost of constructing the portfolio. Next, suppose we are given a collection of T distinct coupon bond prices. Then, each coupon bond has an equation similar to Equation 1. This gives us T equations in the T unknowns {B(0,1), B(0,2), . . . , B(0, T)}. One can solve this system of linear equations using linear algebra to obtain the zero-coupon bond prices. If the system does not have a solution, one can find the zero-coupon bond prices that minimize the sum of errors squared between the right side of Equation 1 and the coupon bond prices on the left side. Finally, one can input the zero-coupon bond prices obtained into Equation 21.9 to get the forward rates.

EXT. 21.3 EX. 1: Computing Forward Rates from Coupon Bond Prices ■

Today is time 0. Consider three US Treasury securities trading with par value L = $100. - The one-year Treasury bill has a price of P1 = $97. - The two-year Treasury note has a price P2 = $100.60 and pays a coupon C2 = $4 annually. - The three-year Treasury note has a price P3 = $101.90 and pays a coupon C3 = $5 annually.

Computing Zero-Coupon Bond Prices ■

Fortunately, the first T-bill gives the zero’s price. Equation 1 gives P1 = LB (0, 1) 97 = 100B (0, 1) or B (0, 1) = $0.97

(2a)

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Using the two-year T-note, Equation 1 gives P2 = C2 B (0, 1) + (C2 + L) B (0, 2) 100.60 = 4 × 0.97 + (4 + 100) B (0, 2) or B (0, 2) = (100.60 − 3.88) /104 = $0.93

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(2b)

Finally, for the three-year T-note, Equation 1 gives P3 = C3 B (0, 1) + C3 B (0, 2) + (C3 + L) B (0, 3) 101.9 = 5 × 0.97 + 5 × 0.93 + (5 + 100) B (0, 3) or B (0, 3) = (101.9 − 4.85 − 4.65) /105 = $0.88

(2c)

Computing Forward Rates As these zero-coupon bond prices are the same as the corresponding values in Example 21.4, we have already computed the forward rates and reported them in Table 21.3. Once you have determined the zero-coupon bond prices, you can easily compute their yields and plot them against time to maturity to obtain the zero-coupon bond price curve.

21.5

The Basic Interest Rate Derivatives Contracts

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A Brief History Interest rate derivatives (IRDs) were created during the 1970s when the placid world of sluggish interest rates gave way to an era of unprecedented interest rate volatility. Recall from our discussions in Chapter 8 that this happened because oil shocks and other supply-side disturbances hiked up inflation, making interest rates more volatile than before. Businesses started seeking tools for managing interest rate risks, which generated innovations in exchange-traded IRDs: ■

In 1975, the Chicago Board of Trade (CBOT) launched the first IRD, the GNMA (Government National Mortgage Association) futures, which could be used to hedge long-term interest rate risk.

■

In 1976, thirteen-week Treasury bill futures contracts started trading. Currently these are the oldest exchange-traded interest rate futures contracts.

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During the late 1980s, GNMA futures contracts became obsolete because US Treasury bond futures, which the CBOT introduced in 1977, provided a better hedge for the same risk.

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In 1981, the Chicago Mercantile Exchange (CME) introduced the world’s first cash-settled derivative, Eurodollar futures. It provided a hedge for short-term interest rate risk, and thirteen years later, it replaced the T-bond futures as the world’s most actively traded futures contract.

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

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In 1982, the CBOT introduced options on T-bond futures, whose pricing and hedging are discussed in Chapters 23–25.

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In 1992, the CME launched a post-market global electronic transaction system Globex. It has helped the growth of the IRD market, which has long been a twenty-four-hour market owing to the influence of world events on interest rates.

Diverse IRDs trade in today’s markets. Recall that the basic equity (and commodity) derivatives on an underlying stock (or commodity) are forward contracts, futures contracts, European calls, and European puts. Analogous contracts trade for IRDs. In this case, the underlying “asset” is the spot interest rate, which is an index and not a traded security. The basic derivatives are the same but this time they have different names: forward rate agreements (FRAs), interest rate futures, caplets, and floorlets. Not all of these basic securities trade, however. Traded are over-the-counter FRAs and exchange-traded futures. Caplets and floorlets trade only as parts of portfolios called caps and floors, respectively. Moreover, portfolios of FRAs and zero-coupon bonds also trade. These are interest rate swaps, which were introduced in Chapter 7. This fact is demonstrated in the next chapter. The next section introduces FRAs and interest rate futures contracts. We first present FRAs, which are easy to understand, and related to forward rates. Next follow Eurodollar futures, which are immensely popular and similar to T-bill futures (which are discussed in Extension 21.4).

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Forward Rate Agreements A forward rate agreement is analogous to a forward contract and is sometimes simply known as a “forward.” Just as a forward fixes a price for buying a commodity at a later date, a forward rate agreement fixes an interest rate for borrowing (or lending) over some future time period. The largest market for this over-the-counter contract is in London, where various dealers quote FRA rates for loans denominated in eurodollars, euribor (Euro Interbank Offered Rate, a libor-like interest rate index computed on euro interbank term deposits), pounds sterling, Swiss francs, and Japanese yen, among others.The next example illustrates the workings of an FRA.

EXAMPLE 21.6: A Forward Rate Agreement ■

Today is January 1 (time 0). International Treats and Restaurants Inc.’s (INTRest, a fictitious name) treasurer plans to borrow $100 million five months from today to build a factory. She likes the current interest rate and decides to lock it in by entering into a long position in a forward rate agreement.

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London based Dealerbank Corp. (fictitious name) quotes on a 5 × 8, 100 million Eurodollar FRA a rate of 5.95 percent to 6 percent. Figure 21.6 shows the workings of this FRA (where we have simplified some market conventions related to the settlement date). - The 5 × 8 means that the FRA begins five months from today (June 1) and terminates on the eighth month.

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- The notional principal LN is $100 million. This amount never changes hands but is used for computing interest payments. This is a Eurodollar FRA because the interest rates are computed in terms of the libor index rates for eurodollars. - The treasurer decides to buy the FRA from the dealer at the fFRA rate of 6 percent. This becomes the fixed borrowing rate that the company locks in. Alternatively, if INTRest wants to loan funds, it can sell the FRA for 5.95 percent, which becomes the effective lending rate. ■

Five months later, on June 1, the 3 month Libor rate index is 8 percent per year. The buyer of the FRA collects a payment because this floating three-month libor rate index (known as the reference rate) is higher than the fixed FRA rate contracted in advance.

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The treasurer gets paid on the payment date September 1 (which is the termination date) Notional principal × (Reference rate − fFRA ) × Time period = 100, 000, 000 × (0.08 − 0.06) × (92/360)

(21.13a)

= $511, 111.11 ■

Some FRAs pay on the settlement date, which is the day the reference rate is announced (five months from today in this example). In that case, to get the payment, discount the cash flow in Equation 21.13a

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FIGURE 21.6: Timeline and Cash Flows of an FRA

Now (start date) (Time 0 = January 1)

Settlement date (June 1)

The company buys an FRA on LN = On June 1, the three$100 million with an FRA rate fFRA = month libor rate index 6 percent per year. is 8 percent per year.

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Payment date (September 1) On September 1, the company gets $511,111.11.

using 8 percent, which is the market interest rate for this 92-day period. Then Long’s company collects on June 1: 511, 111.11 = $500, 871.08. (21.13b) 92 (1 + 0.08 × 360 ) INTRest will have to pay 8 percent if it borrows money in June. The effective borrowing cost is only 6 percent because of the payment received on the FRA.

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

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The protection lasts only for three months. To hedge cash flows for a longer time horizon, INTRest can enter into a series of forward rate agreements of different maturities. The company would pay 6 percent and receive the floating reference rate for the relevant periods. Fortunately, this contract trades in the market as a package. As you may guess, this is the interest rate swap contract that we introduced in Chapter 7! The next chapter will formalize this intuition and show why an interest rate swap is a collection of FRAs. Notice the following features of an FRA:

1. The FRA uses the actual/360-day count convention. Interest is computed over the actual number of days in the period, assuming there are 360 days in the year. FRAs on Eurodollars and Euribors follow this convention, while FRAs on pounds sterling follow the actual/365-day count convention. 2. The payment is discounted with the realized libor rate index by using simple interest rates. 3. We have assumed that there is no risk of default on the part of the writer of the FRA, implying that the payment contracted is the payment received. As an FRA is an over-the-counter instrument, collateral is needed to support the transaction, which the counter-parties negotiate. 4. As with a forward contract, the payoff from an FRA may be positive, zero, or negative. Thus, an FRA is a “bet” on the future movements of the libor rate index.

The FRA’s payoff in this example can be formalized. Consider a forward rate agreement with notional principal LN , settlement date tSet , and payment date T. The forward rate agreement’s payoff on the payment date (time T) is

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LN × (Ref − fFRA ) (T − tset ) /360

(21.14a)

where Ref is the reference rate (the libor rate index or Euribor in most cases) determined at time tSet that applies over the time period (T – tSet ) in days and where fFRA is the FRA rate. If the cash flow takes place on the day the reference rate is determined, then the FRA payoff on the settlement date (time tSet ) is LN × (Ref − fFRA ) (T − tset ) /360 [1 + Ref × (T − tset ) /360]

(21.14b)

where the payoff in Equation 21.14a is discounted to the settlement date by the reference rate. In our discrete time model, the FRA payoff at time T simplifies to LN × [R (T − 1) − fFRA ]

(21.14c)

because the reference rate is R(T – 1), the spot rate of interest at time (T – 1), and the time period has unit length (not 360 days).

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Interest Rate Futures As you may guess, an interest rate futures contract is the exchange-traded “fraternal twin” of an FRA. This is because it’s also a contract that fixes a borrowing (or a lending) rate at some future date. It differs from an FRA in the same way that a futures contract differs from a forward. A host of features distinguish them such as standardization, a clearinghouse, margins, and daily settlements. Some interest rate futures (like those on Treasuries) end in physical delivery, whereas others (like those on Eurodollars) end in cash settlement. Futures prices get quoted the same way as the underlying security’s prices are quoted. Consider some highly popular interest rate futures contracts that trade at the CME Group’s exchanges. Futures prices for Treasury bonds, notes, and STRIPS are quoted in 32nds, with $100 as the par value. By going long a futures contract on one of these securities, you are promising to buy the underlying bond at a fixed price at a future date. However, Eurodollar futures, like Treasury bill futures, follow a different quoting convention. We discuss this in the context of Eurodollar futures, which are free from the complications related to the embedded options present in the physical delivery feature of T-bill futures contracts (Treasury futures are discussed in Extension 21.4). Eurodollar futures contracts’ payoffs are based on the three-month libor rate index at the future delivery date of the contract, but instead of directly quoting futures interest rates, Eurodollar futures contracts are quoted in terms of futures prices, which are defined as

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Quoted futures price = 100 − Futures interest rate.

(21.15)

For example, a quoted futures price of 94 implies a futures interest rate of (100 – 94) = 6 percent. Notice the following features of Eurodollar futures. 1. Obviously, dividing the futures interest rate by 100 gives the interest rate as a decimal, 0.06 in this case. The decimal point format is used in subsequent computations. 2. Ninety-four is the quoted futures price for a Eurodollar deposit with $100 par value paid three months after the contract matures. The quoted Eurodollar futures price, a number between 0 and 100, sets the market value of a newly created Eurodollar futures contract to zero. 3. This method of quoting futures prices has yet another benefit: it restores the familiar relation between ask and bid prices seen in other markets. Recall that in most markets the ask price (which is the buying price for the customer, abc) is higher than the bid price (which is the selling price of the customer). The same relationship holds in the interest rate market in term of prices: the ask price paid to buy a bond is higher than the bid price at which it is sold. However, bonds usually get quoted in terms of an interest rate: the yield. Due to an inverse relationship between a bond’s price and its yield, the ask yield rate is lower than the bid yield rate. Similarly, the quoted futures price, as Example 21.7 illustrates, places the quoted ask futures price higher than the quoted bid futures price (see Figure 21.7), and the ask futures interest rate is lower than the bid rate.

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

EXAMPLE 21.7: A Eurodollar Interest Rate Futures Contract

Eurodollar Futures Contract ■

Suppose today is January 1, and the June Eurodollar futures has a quoted ask price of 94.00 and a quoted bid price of 93.99. This means the ask futures interest rate is 100 – 94 = 6 percent per annum, while the bid futures interest rate is 100 – 93.99 = 6.01 percent (see Figure 21.7). The underlying asset is a 90-day 1 million Eurodollar three-month deposit. One million dollars is the notional amount, and the contract is cash-settled. The futures interest rate is always computed over a 90-day period (see Table 21.5 for the CME Group–traded Eurodollar futures contract’s specifications).

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It is important to note that the quoted futures price (94 for ask) is the convention that the market uses to determine the futures interest rate. Here, the ask futures interest rate is 6 percent or ifut = 0.06 in decimal terms.

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The futures interest rate is used to determine the payments to the futures contract. The time t quantity that determines the payments to the futures contract is given by the formula 90 1, 000, 000 × (1 − ifut × (21.16a) 360 )

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where ifut is the futures interest rate expressed as a decimal. For convenience, we call this quantity the futures-contract-price. ■

The difference between the quoted futures price and the futures-contract-price is an adjustment for the notional amount (from 100 to 1 million dollars) and an adjustment for the three-month time period over which the simple interest rate applies (note in Equation 21.15 that the quoted futures price is implicitly for a one-year time period).

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The ask futures-contract-price is 1,000,000 × [1 – 0.0600 × (90/360)] = $985,000, and the bid futures-contract-price is 1,000,000 × [1 – 0.0601 × (90/360)] = $984,975 (see Figure 21.7). Notice that if the quoted futures price goes down by one basis point, or 0.0001, then the Eurodollar futurescontract-price goes down by $25.

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The futures-contract-price on the delivery date is 1, 000, 000 × (1 − i ×

90 360 )

(21.16b)

where i is the Libor spot interest rate index at the delivery date expressed as a simple interest rate. Note that on the delivery date, we get convergence of the futures interest rate to the spot rate. ■

The futures-contract-price on the delivery date represents the value on that date of a Eurodollar deposit paying a million dollars in three months time. This computation uses a simple interest rate based on the 360 day year convention. For example, if i = 0.03 on the delivery date, then the futures contract price is 1,000,000 × [1 – 0.03 × (90/360)] = $992,500. This is the purchase price on the delivery date of a Eurodollar deposit that pays one million dollars after three months.

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FIGURE 21.7: Futures Price, Futures Rate, and FuturesContract-Price for Eurodollar Futures Ask futures-contract-price = $985,000 Ask futures price = $94 Bid futures rate = 6.01 percent

Ask futures rate = 6 percent Bid futures price = $93.99 Bid futures-contract-price = $984,975

Hedging Example

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■

Let us tailor this contract to meet INTRest’s hedging needs as enumerated in the previous example: - INTRest needs one hundred contracts to match the $100 million. - INTRest must sell Eurodollar interest rate futures contracts to fix the borrowing cost: if the interest rate increases, the futures price declines, the futures generates a negative payment, and being short INTRest receives a positive cash payment. - For example, if the futures price for the ask declines to 91.99 (which corresponds to a futures interest rate of 100 – 91.99 = 8.01 percent), then the new futures-contract-price is 1,000,000 × [1 – 0.0801 × (90/360)] = $979,975.

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Being short one hundred contracts, INTRest has a cash flow of −100 × [New contract value (for ask) − Old contract value (for bid)] = −100 × (979, 975 − 984, 975) = $500, 000 The contract’s performance is similar to that of the FRA contract (see payoff in Equation 21.13b).

Margins ■

We have considered the bid/ask spread but neglected the other transactions costs (such as brokerage commissions) as well interest earned or paid on the margin account. Margin requirements for Eurodollar futures are complex. They depend on the trading strategy as well as other factors, and speculators have higher margin requirements than do hedgers or exchange members.

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January 1 - Ms. Long buys one June Eurodollar contract at a futures price of 94. As noted earlier, this corresponds to a futures-contract-price of $985,000.

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

- No money changes hands today. Long keeps $1,000 initial margin with her broker. Her maintenance margin is $750 per contract. ■

January 2 - Settlement for the futures price is 94.05. The futures interest rate is 100 – 94.05 = 5.95 percent or 0.0595 as a decimal. The futures-contract-price is 1,000,000 × [1 – 0.0595 × (90/360)] = $985,125. - The margin account adjustment = Futures-contract-price (Jan 2) – Futures-contract-price (Jan 1) = 985,125 – 985,000 = $125. Her margin account balance is 1,000 + 125 = $1,125. - Notice that with the futures price increasing, the long position has gained value.

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January 3 - Settlement for the futures price is 93.75. The futures-contract-price is 1,000,000 × [1 – 0.0625 × (90/360)] = $984,375. - The margin account adjustment = 984,375 – 985,125 = – $750. - As her margin account’s value 1,125 + (–750) = $375 has fallen below the maintenance margin level, she gets a margin call from her broker to deposit (1,000 – 375) = $625 or more in cash.

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TABLE 21.5: Eurodollar Futures Contract Specifications Specification

Description

Underlying instrument

Eurodollar time deposit having a principal value of $1 million with a three-month maturity

Tick size (minimum fluctuation)

One-quarter of one basis point (0.0025 = $6.25 per contract) in the nearest expiring contract month; one-half of one basis point (0.005 = $12.50 per contract) in all other contract months

Contract months

March, June, September, December, extending out ten years (total of forty contracts) plus the four nearest serial expirations (months that are not in the March quarterly cycle). The new contract month terminating ten years in the future is listed on the Tuesday following expiration of the front quarterly contract month

Last trading day

The second London bank business day prior to the third Wednesday of the contract expiry month; trading in the expiring contract closes at 11:00 am London time on the last trading day (Continued )

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TABLE 21.5: (Continued ) Specification

Description

Final settlement

Expiring contracts are cash settled to 100 minus the three-month US dollar libor rate index on the last trading day; final settlement price will be rounded to four decimal places, equal to 1/10,000 of a percent or $0.25 per contract

Position limits

None

Trading hours (Central Time)

Open outcry, Monday–Friday: 7:20 am – 2:00 pm; CME GLOBEX, Sunday–Friday: 5:00 pm – 4:00 pm

Source: www. cmegroup. com/trading/interest-rates/stir/eurodollar_contract_specifications.html

The Eurodollar’s futures-contract-price in the preceding example can be formalized. Consider a Eurodollar futures contract at time t with delivery at time T and notional value LN = $1 million on a three-month Eurodollar deposit. The futurescontract-price of the Eurodollar futures at time t is given by the formula

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F (t) = LN × [1 − ifut (t) ×

90 360 ]

(21.16c)

where F(t) is the time t futures-contract-price and ifut (t) is the time t futures interest rate expressed as a decimal. Note that the quoted futures price does not explicitly appear in this equation. The futures price is just the convention used by the market to quote the futures interest rate. Marking to market over the time period [t, t + 1] is given by the change in the futures-contract-prices, that is F (t + 1) − F (t) = −LN × [ifut (t + 1) − ifut (t)] ×

90 . 360

(21.16d)

As the futures interest rate increases, the futures-contract-price declines, as indicated by the minus sign in this equation. This is intuitive because futures-contract-prices and futures interest rates move inversely to each other. The settlement futures-contract-price for the futures contract on the delivery date is 90 (21.16e) 360 ] where F(T) is the time T futures-contract-price and i(T) is the 90-day libor rate index spot rate at time T expressed as a simple interest rate. As we approach the delivery date, note that we get convergence of the futures interest rate ifut (t) to the spot rate of interest; that is, as t goes to T, ifut (t) approaches i(T). F (T) = LN × [1 − i (T) ×

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

529

In our discrete time model, setting the notional value of the Eurodollar futures contract to a dollar, the futures-contract-prices at time t and the delivery date time T simplify to F (t) = [1 − ifut (t)]

(21.16f)

F (T) = [1 − R (T)]

(21.16g)

and because the time period has unit length, and the spot rate of interest (see Equation 21.12) is also the simple interest in this case. We note that in this discrete time case, the quoted futures price is the futurescontract-price expressed as a percentage (multiplied by 100). For our subsequent modeling, therefore, the distinction between the quoted futures price and the futurescontract-price is no longer needed.

EXTENSION 21.4: Treasury Futures

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Treasury Bill Futures Price Introduced in 1976, the 13-week Treasury bill futures contract is the oldest exchange-traded interest rate derivative. A highly popular contract for hedging and speculating on short-term interest rate risk, it shares many common features with the Eurodollar futures contract: both contracts are written on a notional principal of $1 million, both assume that interest applies over 90 days, both use 360 days in a year, and both are structured so that a one–basis-point movement in the index translates into a $25 change in the futures-contract-price. The key difference is that a T-bill futures contract has a T-bill as the underlying asset, which must be delivered physically if the short delivers, while a Eurodollar futures contract is cash-settled with no future delivery requirement. To see how a 13-week T-bill futures works, let us start with an ask futures price of $94.00. Similar to the Eurodollar futures (see Example 21.7), this translates into an interest rate of 100 – 94 = 6 percent per year. This is the banker’s discount yield that you saw in Equation 2.7b. Using this interest rate as the ask, Equation 2.7b gives an ask futures-contract-price for the bill with a face value of $1 million and 90 days to maturity: T PAsk = 1, 000, 000B = 1, 000, 000 1 − (banker’s discount yield) [ ( 360 )] 90 = 1, 000, 000 (1 − 0.06 × 360 ) = 985, 000 If the quoted futures price is $94.01, then the banker’s discount yield is 5.99 percent, and the bond’s ask futurescontract-price is $985,025. Similar to the Eurodollar futures, a rise in the futures price by one basis point is associated with a $25 increase in the 13-week Treasury bill futures-contract-price, a gain that will accrue to the buyer of the T-bill futures. The physical delivery requirement makes the contract more cumbersome than Eurodollar futures. At the maturity of the T-bill futures, a newly issued T-bill becomes available with 13 weeks to maturity. The CME Group allows the substitution of 91-day and 92-day T-bills with appropriate price adjustments to enhance the liquidity of the deliverable security.

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Treasury Note and Bond Futures Longer-maturity Treasury futures work similar to T-bill futures, but they are quoted differently. Treasury bonds, notes, and STRIPS are quoted in 32nds with $100 as the par value. By going long a futures contract on one of these securities, you are promising to buy the underlying bond at a fixed price at a future date. As with T-bill futures, the short initiates the delivery and decides on which bond to deliver. The contracts allow substitution among several possible bonds with appropriate price adjustment via conversion factors. This gives the short a valuable embedded option in the form of providing the cheapest-to-deliver bond. Moreover, the short has another “option,” known as the wild card option, because the T-bond futures stop trading at 2:00 pm. (Central Time), while bonds trade until 4:00 pm and the short has until 8:00 pm to issue the notice of intention to deliver. These embedded options significantly complicate the valuation of Treasury futures contracts.

The Equivalence between Forward and FRA Rates Both forward rates and forward rate agreements lock in an interest rate for borrowing or lending starting at some future date. Is there a link between the two? The answer is yes, they are equal. Let us illustrate this with a simple example (see Example 21.8).

EXAMPLE 21.8: Forward and FRA Rates

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Using the assumption of no arbitrage, we will determine the FRA rate and demonstrate that it is equivalent to the forward rate. - Consider a forward rate agreement written at time 0 that starts at time 3 and matures at time 4. - Denote the FRA rate by fFRA (0,3) ≡ fFRA . The contract is on a notional principal of LN = $100 million. - The FRA contract is written on the spot interest rate’s value at time 3, R(3). This spot rate becomes known at time 3 and is effective over the time interval [3,4] (see Equation 21.14c).

At the FRA’s Maturity Date, Time 4 ■

The cash payoff at the FRA’s maturity date is [R (3) − fFRA ] 100 = [1 + R (3) − (1 + fFRA )] 100

(21.17a)

where we rewrite the payoff by adding and subtracting 1.

At Time 3 ■

Moving back one period and computing the time 3 present value gives [1 − (1 + fFRA ) B (3, 4)] 100.

(21.17b)

THE BASIC INTEREST RATE DERIVATIVES CONTRACTS

The first term is 1 because we get [1 + R(3)] at time 4 by investing a dollar in a money market account at time 3, and the second term is obtained by multiplying (1 + fFRA ) at time 4 by the zero-coupon bond price B(3,4) to get its present value at time 3.

At Time 0, When the FRA is Negotiated ■

To compute the time 0 present value, note two facts (see Figure 21.1): the time 0 present value of a dollar at time 3 is B(0,3); similarly, the time 0 present value of B(3,4) is B(0,4). The present value of the preceding cash flow is therefore [B (0, 3) − (1 + fFRA ) B (0, 4)] 100

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(21.17c)

As with a conventional forward contract, the FRA rate sets the value of the FRA to zero on the day it is written (time 0). Equating Equation 21.17c to zero and solving for the FRA rate gives: [B (0, 3) − (1 + fFRA ) B (0, 4)] 100 = 0 or fFRA (0, 3) ≡ fFRA = [(B (0, 3) /B (0, 4)] − 1 ≡ f (0, 3)

(21.17d)

where the last term follows from the definition of the forward rate as given in Equation 21.9.

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The argument in the previous example works more generally. Replacing time 4 by T in Equation 21.17d yields the following result (see Result 21.3).

RESULT 21.3 The FRA Rate and Forward Rates Consider an FRA with notional LN , maturity date T, and settlement date T – 1 in the discrete time model. Then, the FRA rate, fFRA (0, T − 1) = f (0, T − 1) , the forward rate

(21.18)

This result is just as it should be. Recall from the previous section that the forward rate with maturity (T – 1) is that rate one can contract at time 0 for riskless borrowing and lending over [T – 1, T]. In contracting, the borrower is betting that the spot rate of interest will be greater than the forward rate—but this is the same bet made when going long an FRA. When going long an FRA, one is betting that the reference rate, here the spot rate of interest, is greater than the FRA rate. Therefore, the two rates must be the same. This is a powerful insight: the abstract forward rate that we had defined earlier is the same as the interest rate that emanates from the market-traded forward rate agreement contract. The power of forward rates is further illustrated in the context of other derivatives in the next three chapters.

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21.6

Summary

1. The discussion of interest rate derivatives (IRDs) parallels the earlier presentation of spots, forwards, futures, and options on commodities and equities. For IRDs, an array of different maturity default-free zero-coupon bonds replace the single commodity or stock price as the underlying asset. Yields and forward rates extracted from the zeros are foundational concepts. We focus on Eurodollar IRDs, which are simpler than derivatives on Treasury securities and attract the greatest trading volume. 2. Eurodollars are dollar deposits in European banks (or a European subsidiary of a US bank) that earn interest in dollars. There are three major interest rates for Eurodollars: the Eurodollar interest rate paid on Eurodollar deposits, the generic libor at which London banks offer to lend dollars to each other, and a libor rate index (which is an average of libor quotes collected from major London-based banks. 3. Assuming a discrete framework in which time progresses in unit increments t = 0, 1, 2, 3, . . . , the yield (or yield to maturity or internal rate of return) on a zero-coupon bond is the interest rate that equates the discounted bond’s cash flows to its current price: B (0, T) =

1 [1 + R(0, T)]T

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where B(0, T) is the time t price of a zero-coupon bond that pays a dollar at time T, and R(0, T) is the yield at time t on a zero-coupon bond with maturity T. 4. Standing at time t, the yield on the shortest-maturity zero-coupon bond that matures at time (t + 1) has a special name—the spot rate of interest R(t). A riskless money market account earns the spot rate of interest each period. 5. The yield curve plots the zero-coupon bonds’ yields as a function of each bond’s time to maturity. This graph is called the term structure of interest rates. A yield curve can take various shapes influenced by factors such as (1) the expectations of the market participants regarding the future movement of interest rates, (2) risk aversion, and (3) supply and demand considerations for the underlying zerocoupon bonds. 6. A default-free coupon bond’s yield (or yield-to-maturity, also called the internal rate of return) y is defined by PT =

C C C C+L + +…+ + 2 T−1 (1 + y) (1 + y) (1 + y)T (1 + y) T

=

C L + ∑ [ (1 + y)t ] (1 + y)T t =1

where PT denotes the coupon bond price, C denotes the yearly coupon, L denotes principal (or par or face value), and T denotes time to maturity in years.

SUMMARY

If the yield equals the coupon rate, the bond’s price is identical to the par value, and the yield is called the par bond yield. 7. Traditional interest rate management involves computing Macaulay’s duration and using it to approximate a bond’s price change for a given change in the yield. Result 21.1 states that for a small change in the bond’s yield to maturity y, the change in the bond’s price is approximately ΔP ≈ −P DurM Δy where P is the original bond price, ∆P is the change in bond price, DurM ≡ Dur/ (1 + y) is the modified duration, and ∆y is the change in the yield-to-maturity. The duration is computed by T

1 C L Dur = ×T t ×t+[ P {∑ (1 + y)T ] } t =1 [ (1 + y) ]

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where C is yearly coupon, L is the bond’s par (or principal) value, T is the bond’s maturity, and y is the original yield to maturity. The price approximation can be improved by a convexity adjustment, which uses the second-order term in a Taylor series expansion for the price change formula. 8. Duration is a weighted average of the time periods to the receipt of the cash flows. A bond with higher duration undergoes a greater price change given a change in the interest rate. The duration of a zero-coupon bond is the same as the time to maturity. A bond portfolio’s duration is a weighted average of the durations of the individual bonds. 9. Duration-based hedging has many shortcomings that limit its efficacy. It only works for small changes in interest rates and for parallel shifts of the yield curve, which means that all yields change by the same amount. The yield curve rarely changes by parallel shifts. 10. The time 0 forward rate for the future period [T – 1, T] is defined as f (0, T − 1) =

B (0, T − 1) −1 B (0, T)

where B(0, T– 1) and B(0, T) are today’s zero-coupon bond prices for bonds maturing at times (T – 1) and T, respectively. 11. The zero-coupon bond price and yield can be expressed in terms of forward rates as B (0, T) =

1 [1 + f (0, 0)] [1 + f (0, 1)] … [1 + f (0, T − 1)]

1

R (0, T) = ([1 + f(0, 0)][1 + f(0, 1)] … [1 + f(0, T − 1)]) T − 1.

533

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CHAPTER 21: YIELDS AND FORWARD RATES

where B(0, T) is the time 0 price of a zero-coupon bond maturing at time T, R(0, T) is the zero-coupon bond’s yield to maturity, and f (0, t) represents the forward rate contracted at time 0 for time t. 12. The spot rate of interest (denoted by R[t]) is also the forward rate for immediate borrowing and lending for one period, that is, R(t) = f (t,t). 13. A forward rate agreement (FRA) is an over-the-counter contract that fixes the interest rate for borrowing (or lending) over some future time period. It has several features: (1) an FRA uses the actual/360-day count convention, (2) the payment is discounted with the realized libor rate index using simple interest, and (3) like a forward, the payoff from an FRA can be positive, zero, or negative. The FRA rate is the same as the forward rate. 14. Consider an FRA with notional principal LN , settlement date tSet , and payment date T. The payoff on the payment date is LN × (Ref − fFRA ) (T − tSet ) /360 where Ref is the reference rate (the libor rate index or euribor in most cases) determined at time tDec that applies over the time period (T – tSet ) in days, and where fFRA is the FRA rate.

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15. An interest rate futures is an exchange-traded contract that fixes a borrowing (or a lending) rate at some future date. It differs from an FRA in the same way as a futures contract differs from a forward. A host of features distinguish them such as standardization, a clearinghouse, margins, and daily settlements. 16. Eurodollar futures prices, like Treasury bill futures, get quoted in terms of an index, which is defined as Futures price = 100 − Futures interest rate 17. The asset underlying a Eurodollars futures contract is a 90-day 1 million Eurodollar three-month deposit. The futures interest rate is always computed over a 90-day period, and the contract is cash-settled. The Eurodollar futurescontract-price at time t, which determines the payments to the futures contract, is given by the formula 1, 000, 000 × [1 − ifut ×

90 360 ]

where ifut is the futures interest rate obtained from the preceding formula for the futures price expressed as a decimal. 18. Consider an FRA with notional LN , maturity date T, and settlement date T – 1 in our discrete time model. Then, the FRA rate, fFRA (0, T – 1) = f (0, T – 1), the forward rate.

QUESTIONS AND PROBLEMS

21.7

Cases

Pension Plan of Bethlehem Steel—2001 (Harvard

Business School Case #202088-PDF-ENG). The case analyzes the company’s pension fund assets and liabilities by using the traditional tools of interest rate risk management.

Union Carbide Corp.: Interest Rate Risk Management (Harvard Business School

Case #294057-PDF-ENG). The case studies how the firm can manage its exposure to interest rates by matching the duration of its liabilities to the duration of its assets. Deutsche Bank: Finding Relative Value Trades (Harvard Business School Case

#205059-PDF-ENG). The case considers how the bank can find yield curve trades that are of interest to clients as well as to its proprietary trading desk.

21.8

Questions and Problems

21.1. What is the difference between a Eurodollar, a Eurodollar rate, and the US

T-bill rate? 21.2. What is libor? Explain how it is similar or not similar to a libor rate index. 21.3. Compute the yield curve for the following zero-coupon bond prices.

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Times to Maturity T

Today’s Zero-Coupon Bond Prices B(0, T)

1

B(0, 1)

0.98

2

B(0, 2)

0.96

3

B(0, 3)

0.94

4

B(0, 4)

0.90

21.4. How does the bond’s yield relate to an internal rate of return? Is the bond’s

yield a good measure for its expected return over the next year? The next three questions use the following information. Bondy Bond (fictitious name) is a coupon bond that matures in 10 years. The coupon rate is 5 percent per year. The bond’s principal is $10,000 and the market price is $8,500. 21.5. (Microsoft Excel) Compute the yield for Bondy Bond, showing how you use

the Excel program for this purpose. 21.6. Compute Bondy Bond’s duration using the yield computed in the previous

problem. 21.7. Suppose the yield increases by 0.0005 over a week. a. Compute the new price of Bondy Bond.

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CHAPTER 21: YIELDS AND FORWARD RATES

b. Compute the actual change in bond prices. c. Use a duration based formula to predict the change in Bondy Bond’s price.

How does this compare with the actual change? 21.8. A coupon bond has a life of 5 years, makes semiannual coupon payments at

the rate of 6 percent per year, has a principal of $100, and a market price of $95. a. Using Excel, compute this bond’s yield to maturity. b. Compute the duration of the bond. 21.9. At time 0, what is the duration of a zero-coupon bond with maturity T?

Prove your answer using the definition of duration. 21.10. What are the two problems with using duration as a risk measure for changes

in bond prices? 21.11. Compute the forward rates for the following zero-coupon bond prices.

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Times to Maturity T

Today’s Zero-Coupon Bond Prices B(0, T)

1

B(0, 1)

0.98

2

B(0, 2)

0.96

3

B(0, 3)

0.94

4

B(0, 4)

0.90

21.12. Compute the forward rates and yields for the following zero-coupon bond

prices. Times to Maturity T

Today’s Zero-Coupon Bond Prices B(0, T)

1

B(0, 1)

0.93

2

B(0, 2)

0.90

3

B(0, 3)

0.88

4

B(0, 4)

0.85

21.13. Using a payoff table similar to Table 21.4 in the text, explain the portfolio of

zero-coupon bonds that generates the forward rate f (0,3). 21.14. Compute the zero-coupon bond prices from the following forward rates.

QUESTIONS AND PROBLEMS

Times to Maturity T

Today’s forward Rates f (0, T)

1

f (0, 1)

0.02

2

f (0, 2)

0.03

3

f (0, 3)

0.04

4

f (0, 4)

0.05

21.15. What is the spot rate of interest R(t)? Explain how it relates to forward rates

and yields. 21.16. What is a forward rate agreement? Give a numerical example to demonstrate

the timing and magnitude of the payoffs to an FRA. 21.17. What is an interest rate futures? Give examples of two popular interest rate

futures. 21.18. What are the differences between a forward rate agreement and an interest rate

futures contract? In your answer, consider both the institutional differences and the economic differences. 21.19. Given the difference between forward rate agreements and interest rate futures

contracts, would you expect the FRA rate to equal the futures interest rate, for otherwise identical contracts? Explain your answer. 21.20. What is the relation between an FRA rate for a contract that matures at time Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

T and a forward rate of interest for time T – 1?

537

22 Interest Rate Swaps 22.1 Introduction

22.4 Valuation

22.2 A Brief History

22.5 Variations of Interest Rate Swaps

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The Introduction of Swap Contracts

22.6 Swaps and FRAs

From a Brokerage to a Dealership Market

Synthesizing Swaps with Eurodollars and FRAs

ISDA and Standardization of Contracts

The Yield and Swap Curve

22.3 Institutional Features Entering a Swap Contract Documentation Closing a Swap Position

EXTENSION 22.1 Computing Forward Rates from Swap Rates

22.7 Summary 22.8 Cases 22.9 Questions and Problems

A BRIEF HISTORY

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22.1

Introduction

“Trust you couldn’t; gamble you shouldn’t.” This characterizes the lessons learned from a fixed-for-floating interest rate swap deal struck in 1993 between Bankers Trust and Procter & Gamble (P& G). Enticed by the fact that the swap would pay a low floating rate, P& G apparently downplayed the importance of a “spread” in its payment terms, which could potentially increase its liability. Interest rates rose, the company suffered huge losses, and it sued, arguing it didn’t understand the complex swap. Suffering a serious erosion of trust, the bank eventually forgave most of the money due and settled the suit. The moral of this story is clear: don’t transact in derivative securities that you do not understand, especially those with complicated terms like an exotic interest rate swap! Understanding such swaps is a purpose of this chapter. Although you have seen swaps before in Chapter 7, we discuss them here in greater detail. The chapter begins with a brief history of swaps. Next, we discuss institutional features of swap contracts. Swaps enjoy great flexibility because they trade in the overthe-counter markets, but the trades are guided by standardized rules. We then develop a valuation formula for plain vanilla interest rate swaps. Last, we demonstrate how to construct swap curves, and we link swaps to forward rate agreements (FRAs) and forward rates. The swap valuation formula presented in this chapter is obtained in a world with stochastic interest rates. But the valuation formula is derived without specifying a particular model for the evolution of the term structure of interest rates. This is due to the linear structure of the swap’s payoffs in both the underlying fixed and floating rate cash flows. The swap’s value is shown to only depend on the current term structure of zero-coupon bond prices. This is analogous to the option pricing relations obtained in Chapter 16, prior to the study of the binomial option pricing model of Chapters 17 and 18 or the Black–Scholes–Merton model of Chapters 19 and 20. If you recall, the nonlinear payoffs of call and put options required the specification of a stock price process to obtain valuations. The same is true for the more general interest rate derivatives discussed in the chapters that follow. The nonlinear payoff structure of interest rate derivatives require the specification of an evolution for the term structure of interest rates, a key element in understanding the Heath–Jarrow–Morton (HJM) model.

22.2

A Brief History

In 1981, the Wall Street firm Salomon Brothers Inc. organized a transaction between the World Bank (often referred to as “the Bank”) and the International Business Machines Corporation (IBM) hailed as the first swap. The World Bank found that interest rates in West Germany and Switzerland were lower than in the United States, but the bank had already borrowed the maximum amount it could in these markets. IBM had loans denominated in Deutsche marks and Swiss francs. An appreciation of the dollar lowered the value of these loans, and IBM wanted to convert them into dollar loans. So, the Bank borrowed dollars by selling Eurobonds and converted them into marks and francs in the spot market. The deal was that the World Bank and IBM

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CHAPTER 22: INTEREST RATE SWAPS

embraced each other’s loan obligations. The net result was the first swap because it involved an exchange of cash flows at future dates. The swap market has enjoyed phenomenal growth—Table 22.1 lists some milestones. We classify the market developments into four categories: (1) the introduction of new swap contracts, (2) the shift from a brokerage business to a dealership market, (3) the founding of the International Swaps and Derivatives Association (ISDA) and the standardization of contracts, and (4) the use of technology and the automation of trading.

The Introduction of Swap Contracts

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The year 1982 saw a financial arrangement usually recognized as the first interest rate swap (see Johnson 2004; Greenbaum and Thakor 2007). The first interest rate swap cleverly matched two parties—Sallie Mae issued an intermediate fixed-rate loan, ITT Corporation issued a floating-rate note—and they swapped their payment obligations. This deal opened the flood gates with respect to swap creation and issuance. Besides developing exotic versions of interest rate and currency swaps, in 1986, Chase Manhattan Bank intermediated the first commodity swap based on oil prices; in 1989, Bankers Trust organized the first equity swap; and in 1995, J. P. Morgan issued the first credit default swap. Swap markets have enjoyed enormous growth. Interest rate and currency swaps were actively traded almost 15 years before either the credit default swap or equity derivatives markets traded significant notional quantities, which happened in 2001 and 2002, respectively.

TABLE 22.1: Some Milestones in the History of Swaps Year

Development

1981

The first currency swap was arranged by Salomon Brothers between IBM and the World Bank.

1982

The first interest rate swap was arranged between Sallie Mae and ITT Financial.

1985

The International Swap Dealers Association (ISDA; later renamed the International Swaps and Derivatives Association) was chartered. ISDA introduced SWAPS, a document template useful for standardizing swap contracts.

1986

The first commodity swap was intermediated by Chase Manhattan Bank.

1987

ISDA published the first Master Agreement, newer versions of which were introduced in 1992 and 2002.

1989

The first equity-linked swap was organized by Bankers Trust.

1995

The first credit default swap was traded by J. P. Morgan.

A BRIEF HISTORY

From a Brokerage to a Dealership Market By 1983, investment banks were arranging one-off swap transactions for their clients on a fully matched basis. Banks earned substantial brokerage fees and had no credit risk—but there were road bumps along the way. Every transaction had to be separately negotiated with the counterparties and carefully documented. The front office (traders) wanted to proceed but the back office (accountants) moved at a snail’s pace, and the trader’s numbers would sometimes differ from the accountant’s. In a particularly telling incident, acute back-office problems forced J. P. Morgan to scale down its swap business for about six months in 1987. The swap banks (also known as swap facilitators) discovered that by becoming swap dealers, they could cut their back-office costs by standardizing the contracts and trading them like homogeneous commodities. The swap banks also started warehousing swaps. They would enter into different swaps on both the buy and the sell side of the market and hedge the residual risk with interest rate derivatives. The next example illustrates how this works.

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EXAMPLE 22.1: Managing a Swap Book ■

Suppose that Swapbank Inc. (a fictitious name) enters into a plain vanilla interest rate swap on a notional principal of $100 million, where it receives the floating-rate six-month libor index rate and pays a fixed rate per year. Like any other good dealer, Swapbank tries to manage its books by perfectly hedging its portfolio.

■

The bank enters into an offsetting swap with matching terms, but it receives four basis points higher interest on a notional principal of $80 million. The bank has hedged its interest rate risk on $80 million and earns four additional basis points, which equals Principal amount × Spread × Time period over which interest is earned = 80 million × 0.0004 × 0.5 = $16, 000 every six months

■

Swapbank has counterparty credit risk. However, given the collateral agreements in the swap market, this risk is small.

■

There is still risk from the remaining $20 million notional principal. Swapbank can enter into another offsetting swap to remove this risk. Swapbank can also remove this risk by trading other interest rate derivatives. We explain how to do this at the end of this chapter.

ISDA and Standardization of Contracts As mentioned previously, to trade swaps quickly and to minimize transaction costs, standardized contracts were needed. Efforts at developing such documents began in 1984, when eighteen swap dealers joined together to form the International Swap Dealers Association (ISDA, pronounced “iz-da”; later renamed the International Swaps

541

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CHAPTER 22: INTEREST RATE SWAPS

and Derivatives Association to reflect its broader scope). Chartered in 1985, ISDA published the Code of SWAPS (Code of Standard Wording, Assumptions, and Provisions for Swaps; the “Code”), which provides standardized documents known as the Master Agreements for over-the-counter (OTC) derivatives contracts. The ISDA’s Master Agreement establishes contract standards for privately negotiated derivatives so that legal uncertainty is reduced and credit risk is lowered through collateral provisions, netting arrangements, and procedures to follow in the case of a default. The Master Agreement itself has two parts: (1) a preprinted form that cannot be amended and (2) a schedule that allows the two counterparties to note changes and amend provisions on the printed form (see Allen and Overy 2002). The Master Agreement establishes noneconomic terms (such as “representations and warranties, events of default, and termination events”) that govern future OTC derivatives transactions between the two counterparties. Each time these two counterparties execute a future trade, all they need do is negotiate the economic terms and record them in a Confirmation. The Master Agreement, the confirmations, and other supporting documents make up the ISDA Agreement Structure (as of 2006).

22.3

Institutional Features

It is instructive to follow a swap transaction from beginning to end and discuss the relevant institutional features met along the way (see Figure 22.1 for the timeline).

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Entering a Swap Contract Suppose Your Beloved Machines (YBM) decides to become a swap issuer (or a swap initiator). Then YBM contacts one or more swap providers who stand ready to be their counterparty. Of course, in selecting the swap bank, YBM needs to consider their credit rating, their prior experience with similar trading, the documentation they provide, and the price they quote. There are three basic ways to initiate a swap contract: 1. The traditional way is by phone, during which the basic economic terms of the transaction (price or rate, principal amount, term, and frequency of payment) are negotiated and any credit support requirements (margin, or collateral, or a guarantee) are discussed. The counterparties are committed to transact as per this “oral agreement.” Then, one counterparty (or broker, if involved) sends a letter containing the economic terms to the other and asks for a confirmation. 2. A second way is via a web-based trading platform, which displays a range of potential swaps. Once YBM has prearranged trading agreements with a bank, the company can check quotes and transact a plain vanilla swap in a matter of minutes. The bank has already made YBM sign the necessary paperwork and set up a system that allows it to trade quickly but safely. 3. A third way is by running an auction. This is more common in the case of municipal swaps that have a state, local, or municipal government as a counterparty.

INSTITUTIONAL FEATURES

FIGURE 22.1: Timeline for a Plain Vanilla Swap

Periodic net payments (dark boxes)

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… Starting date

Intermediate dates

Swap initiator and bank execute trade by verbal agreement, an electronic trade, or an auction. Necessary documentation follows. Unless it is a forex or a currency swap, notional principal is not exchanged.

Counterparties make periodic payments (semi-annually for most swaps).

No cash changes hands (a par swap).

Traders can close out positions by (1) a buyout, (2) an assignment, or (3) an offset.

Counterparties may have to keep collateral.

No secondary market. OTC contract—credit risk is present.

…

…

Maturity date Swap ends with a final payment.

Payments are netted (if they are in the same currency); usually not netted if they are in different currencies.

Documentation “Partnerships often finish in quarrels” wrote Benjamin Franklin in his 1791 autobiography, “but I was happy in this, that mine were all carried on and ended amicably, owing, I think, a good deal to the precaution of having very explicitly settled, in our articles, every thing to be done by or expected from each partner [emphasis added], so that there was nothing to dispute, which precaution I would therefore recommend to all who enter into partnerships.” This practical advice on partnerships, and by extension swaps, is the reason for documentation. The swap contract’s documentation must describe (1) the currencies of the cash flows, (2) the interest rates, (3) the principal amount, (4) whether the principal is

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CHAPTER 22: INTEREST RATE SWAPS

exchanged, and (5) whether the counterparties make future cash payments in their entirety or through a net payment. Next comes a timetable: (6) the tenor (or maturity) of the swap and (7) the frequency of cash payments (every six months for a majority of simple swaps). Finally, the contract must specify other issues such as (8) how to terminate a swap early and (9) what to do if one side defaults. Of course, webbased trading platforms have most of this material prepackaged so that trades can be executed quickly.

Closing a Swap Position If all contract terms are fulfilled, a swap terminates at its maturity date. But there are three ways of ending a swap before it matures. Suppose that YBM decides to close out an existing swap today and calculations show that the swap has a positive value of $1 million. Then, the process of closing is as follows: 1. Buyout. A buyout involves YBM’s counterparty paying YBM $1 million and then ending the swap by mutual consent.

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2. Assignment. With the counterparty’s approval, YBM can assign the swap to a third party. The third party would pay YBM $1 million, replace YBM in the transaction, embrace all the benefits, and accept all the obligations for the swap’s remaining life. 3. Offset. YBM can enter into an identical swap with equal but opposite terms as the original swap. YBM’s counterparty in the new swap will have to pay YBM $1 million because that is the value of the swap. In the combined swap position, YBM will have zero net cash flows in the future. Notice that unlike the first two transactions, here YBM does not completely leave the market and will continue to have credit risk until both swaps mature.

22.4

Valuation

This section values a plain vanilla interest rate swap when interest rates are stochastic. The valuation formula is derived without specifying a particular evolution of the term structure of interest rates. This is due to the linear structure of the swap’s payoffs in both the underlying fixed and floating cash flows. This is analogous to the pricing of forward contracts in Chapters 11 and 12. Swap valuation involves a clever trick: (1) decompose the swap into a fixed rate and a floating rate loan corresponding to each leg of the swap, (2) compute the present value of each of these loans, and (3) take their difference. Of course, we know how to compute the present value of fixed cash flows, but how does one value the floating cash flows? We solve this problem with a simple observation.

VALUATION

The simple observation is that the value of a floating rate loan is always equal to its par value on the payment and reset dates.1 This happens because the loan pays out the interest owed on these dates. For example, a floating rate loan of $100 million for three years pays out the relevant interests perhaps quarterly (by whatever method of interest computation devised) and at each quarterly payment date, the value of the loan outstanding is $100 million. The clever trick in valuing a swap is to include the principal payments while valuing the fixed-rate and floating-rate cash flows. This inclusion does not matter because the principal payments cancel when taking the difference between the two loans’ cash flows. And, it simplifies our task because it is easy to value both the fixed and floating rate loans. To see how this works, consider both a fixed rate loan paying interest for three years and then returning a principal of $100 million, and a three-year floating rate loan with a principal of $100 million. Next, value both of these loans. Now, if you take the difference between these values, the present value of the $100 million notional cancels, leaving just the present value of the fixed-rate payments less the floating rate payments. This is precisely the value of a swap. Armed with these insights, let us value an interest swap.

EXAMPLE 22.2: Valuing a Plain Vanilla Interest Rate Swap (Fixed versus Floating)

A Numerical Example Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

■

1

Consider the data from Example 7.4. - Fixed Towers Inc. borrowed $100 million for three years at a floating rate of one-year libor but now wants a fixed-rate loan. - Floating Cruisers Co. raised the same amount, for the same time period, at a fixed interest rate of i = 0.06 or 6 percent per year and wants to switch to a floating-rate loan. - They do an interest rate swap by embracing each other’s liabilities. Fixed Towers buys the swap by agreeing to pay the fixed rate of 6 percent to Floating Cruisers, who, in turn, sells the swap by agreeing to pay the one-year libor rate to Fixed Towers (see Figure 22.2, which repeats Figure 7.2). A swap buyer pays a fixed rate and receives the floating rate, whereas a seller does the opposite. - The notional principal of LN = $100 million is used to compute interests that are paid at year’s end. - We discount the fixed rate payments using the zero-coupon bond prices in Table 22.2.

It is a simple matter to compute the present value of the next floating interest rate payment between the payment dates. The present value of the next floating interest rate payment plus the par value is the value of the floating rate loan to the issuer between the payment dates. The next floating interest rate payment (say, X) is known at the last repayment date (say, time a) and before it is due (say, time b). The next floating interest rate payment is equal to the current floating interest rate times the notional. Then, the present value at time a of the next floating interest rate payment at time b is B(a, b)X. At time b, when the interest is paid, the floating rate loan resets to par value.

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CHAPTER 22: INTEREST RATE SWAPS

FIGURE 22.2: Plain Vanilla Interest Rate Swap Transforming a Fixed Rate Loan Into a Floating Rate Loan (and Vice Versa) After Entering into Swap for 3 years

Fixed Towers Net cost 6 percent per year

libor 6 percent

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libor

Floating Cruisers Net cost libor per year

6 percent

Present Value of the Fixed Rate Loan ■

First, let us look at the fixed rate loan underlying the swap. The fixed rate loan has a coupon payment of C dollars per period plus a principal of LN at the maturity date. The periodic payments are obtained by multiplying the notional principal by the fixed interest rate i. Therefore The fixed payment at the end of each period, C = LN × i = 100 × 0.06 = $6 million

(22.1)

The present value of the fixed rate loan is PVFix = B (0, 1) × C + B (0, 2) × C + B (0, 3) × C + B (0, 3) × C + B (0, 3) × LN = (0.97 + 0.93 + 0.88) × 6 + 0.88 × 100 = $104.68 million

(22.2)

VALUATION

TABLE 22.2: Common Zero-Coupon Bond Price Data Times to Maturity T

Today’s Zero-Coupon Bond Prices B (0, T)

1

B(0, 1)

0.97

2

B(0, 2)

0.93

3

B(0, 3)

0.88

4

B(0, 4)

0.82

Present Value of the Floating Rate Loan ■

Recall our observation: the floating rate loan is always equal to its par value at the payment dates: PVFloat = LN = $100 million

(22.3)

Swap Valuation

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■

Consequently, the value of the swap to Fixed Towers, who pays fixed and receives floating, is computed by subtracting (22.2) from (22.3): Present value of the floating rate loan − Present value of the fixed rate loan = PVFloat − PVFix = 100 − 104.68 million = −$4.68 million

(22.4)

■

To enter the swap, Fixed Towers needs to be paid $4.68 million today.

■

Alternatively, one can adjust the interest rate to make this a par swap with zero value. To do this, one needs to find the fixed rate that equates the present value of the two legs, that is, LN = B (0, 1) C + B (0, 2) C + B (0, 3) C + B (0, 3) LN Remembering from (22.1) that C = LN × i, rearranging terms, and canceling LN , we get Swap rate iS =

1 − B (0, 3) B (0, 1) + B (0, 2) + B (0, 3)

The fixed interest rate for the par swap, iS , is commonly referred to as the swap rate. ■

Plugging in the values of the zero-coupon bonds, we get iS = (1 − 0.88) / (0.97 + 0.93 + 0.88) = 0.043165 or 4.32 percent (approximately)

This example motivates the following general result.

(22.5)

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CHAPTER 22: INTEREST RATE SWAPS

RESULT 22.1 Valuing a Plain Vanilla Interest Rate Swap Consider an interest rate swap between two counterparties starting today (time 0) and ending at time T. At the end of each period (at time t, where t = 1, 2,.., T), the swap buyer pays a fixed amount LN × i (where LN is the notional principal and i is the fixed interest rate) and receives a floating payment known only at the beginning of that period. The value of the swap to the buyer equals the present value of a floating-rate loan minus the present value of a fixed rate loan, that is, VSwap = PVFloat − PVFix

(22.6)

The present value of the floating rate loan at the payment dates always equals par, that is, PVFloat = LN (22.7) and the value of the fixed rate loan is the present value of the fixed cash flows, that is, PVFix = [B (0, 1) + B (0, 2) + ... + B (0, T)] LN × i + B (0, T) × LN T

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=

B (0, t) LN × i + B (0, T) × LN [∑ ] t =1

(22.8)

where B(0, t) is the time 0 price of a zero-coupon bond that pays $1 at time t. In the case of a par swap, the fixed interest rate iS (commonly known as the swap rate) is computed by setting PVFloat = PVFix .

For simplicity, our previous example assumed annual interest payments. Example 22.3 shows how cash flows are computed in the more realistic case of semiannual payments.

EXAMPLE 22.3: Plain Vanilla Interest Rate Swap with Semiannual Interest Payments ■

Let us continue with the original swap in Example 22.2 but assume that the payments are made every six months. The fixed rate i = 6 percent is quoted on a semiannual bond-equivalent basis. It assumes a 365-day year and computes interest over the actual number of days in this period. If there

VARIATIONS OF INTEREST RATE SWAPS

are 182 days during the next six-month period, Fixed Towers pays after six months: Notional prinicpal × Fixed interest rate × (Days in period/365) = 100, 000, 000 × 0.06 × (182/365) = $2, 991, 780.82 ■

The floating side is quoted on a money market yield basis, which assumes 360 days in a year instead of 365. If the current six-month libor rate is 5 percent per year, Floating Cruisers pays after six months Notional prinicpal × Six − month libor rate × (Days in period/360) = 100, 000, 000 × 0.05 × (182/360) = $2, 527, 777.78

■

Fixed Towers pays Floating Cruisers a net amount equal to 2, 991, 780.82 − 2, 527, 777.78 = $464, 003.04

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■

Every six months, Fixed Towers pays $2,991,780.82 or a similar amount computed using the actual number of days over the previous six-month period. After six months are over, the counterparties check the market and finds the new six-month bbalibor rate that applies over the next six months. With this rate and the actual number of days in the period, the next floating payment is calculated.

22.5

Variations of Interest Rate Swaps

This section explores the different kinds of interest rate swaps that can be created by adding features to plain vanilla interest rate swaps. Interest swaps have become quite sophisticated these days. Here are some examples created by adding various features to plain vanilla interest rate swaps: ■

Setting a maximum value (or a cap) on the floating rate in a plain vanilla swap creates a rate-capped swap. Later you will see that this simple change embeds a call option into the swap. If a minimum value (or a floor) on the floating rate is also specified, then this security embeds an interest rate collar.

■

A yield swap, which has both payments floating, may involve the exchange of a bond with a lower yield for one with a higher yield. Usually the higher-yield bond has a longer maturity and a lower credit rating, which implies a higher interest rate and credit risk.

■

The notional principal in an amortized swap gets reduced over the swap’s life according to a predetermined schedule. This is useful for hedging a portfolio of mortgages, which typically earns a fixed rate on the dollars invested and which amortizes over time owing to principal repayments.

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CHAPTER 22: INTEREST RATE SWAPS

■

There are other swaps that involve changing notional principals. For example, accreting swaps have a notional principal increasing over time. These swaps are useful for hedging the risk associated with a construction project, where more money is invested over time. Seasonal swaps are useful in the retailing business, where the committed capital depends on the time of the year. The most general form of such a swap would be a roller-coaster swap, whose notional principal is structured by the dealer to suit the financing needs of the client’s business.

■

A cancellable swap gives one counterparty the right to cancel the swap without additional payments or penalties. They come in two types. A callable swap gives the fixed-rate payer the right to end the swap and is likely to be called when floating rates decline below a certain level. In contrast, a putable swap gives the fixed-rate receiver the right to put or cancel the swap and is likely to be put when floating rates rise above a threshold level.

■

A forward swap (or a deferred swap) has its terms and conditions set at origination but begins only at a later date. This delaying tactic may be used for accounting or tax purposes.

■

A constant maturity swap (or CMS) involves the exchange of fixed-rate payments for floating rate payments where the floating rate (which is the constant maturity rate) is reset at each period according to a fixed maturity market rate with a duration extending beyond that of the swap’s reset period (such as the one-year constant maturity Treasury note rate).

To value and hedge these many variations of plain vanilla interest rate swaps, one needs the Heath–Jarrow–Morton model to price the embedded options, which is discussed in the next three chapters.

22.6

Swaps and FRAs

The previous chapter noted that forward rates are the key rates underlying the valuation of interest rate derivatives. As the forward rate is identical to the FRA rate, it is not surprising that FRAs can be used to synthetically construct a swap. Let’s see how.

Synthesizing Swaps with Eurodollars and FRAs This section shows how to construct a synthetic swap as a portfolio of FRAs and zeros (Eurodollar deposits). Synthetic construction has several benefits: (1) it provides an alternative way for pricing a swap as the cost of this synthetic construction, (2) it shows how to hedge a swap’s cash flows using FRAs and zeros, and (3) it provides a tool for identifying arbitrage opportunities across the FRA and swap markets. The next example demonstrates this construction.

SWAPS AND FRAS

EXAMPLE 22.4: Synthetic Construction of an Interest Rate Swap with FRAs and Eurodollar Deposits

The Construction ■

A careful examination of a fixed-for-floating interest rate swap’s cash flows reveals how the synthetic construction works. Consider the swap from Example 22.2. - The swap has a notional principal of LN = $100 million. It starts today (time 0) and matures after three years. - The fixed interest rate i is 0.06 per year. - The floating rate is the one-year libor index rate for Eurodollars, R(t). These rates are known at the beginning of each period, that is, R(0) is determined at time 0, R(1) at time 1, and R(2) at time 2.

■

The swap buyer Fixed Towers pays fixed and receives floating. For example, its cash flow is 100 [R (1) − 0.06]

(22.9a)

after two years. Figure 22.3 depicts the buyer’s cash flows. ■

Now, consider an FRA on a notional principal of $100 million that matures after two years. The FRA buyer’s cash flow is given by expression (21.11c) of Chapter 21: 100 [R (1) − fFRA (0, 1)]

(22.9b)

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where fFRA (0,1) is the forward rate negotiated today that starts at the end of the first year and applies over the second year.

FIGURE 22.3: A Swap Buyer’s Cash Flows (Pays Fixed, Receives a Floating Rate)

Time 0

■

1

2

3

Spot R(0) rates

R(1)

R(2)

Cash flows to the Buyer

100[R(0) – 0.06]

100[R(1) – 0.06] 100[R(2) – 0.06]

Except for the last quantity in each expression, the cash flows in expressions (22.9a) and (22.9b) are the same. To adjust for this difference, add to the FRA trade a long position in 100[fFRA (0,1) – 0.06] zero-coupon bonds maturing after two years. This could be a positive or negative quantity, depending

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on whether the spot rate is greater than or less than the FRA rate. The long FRA and the long zeros’ portfolio’s payoff at time 2 will be 100 [R (1) − fFRA (0, 1)] + 100 [fFRA (0, 1) − 0.06] = 100 [R (1) − 0.06]

■

■

(22.9d)

This is the same as the swap’s time 2 cash flow expression (22.9a). As this argument is not specific to time 2, you can synthetically construct each of the swap’s cash flows at the other payment dates in an identical manner. Summing the FRAs and zeros across all the cash flow payment dates generates a portfolio of FRAs and zero-coupon bonds that yield the same cash flows as the swap.

We gather these ideas together as our next result.

RESULT 22.2 Synthetically Constructing a Swap Using FRAs and Zeros

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Consider an interest rate swap starting today (time 0) and ending at time T. At the end of each period t = 1, 2,.., T, the swap buyer pays a fixed amount LN × i (where LN is the notional principal and i is the fixed interest rate) and receives a floating amount, the spot rate R(t − 1). The swap buyer’s cash flows at time t are LN × [R (t − 1) − i]

(22.10)

This swap’s payoffs are identical to a long portfolio of FRAs with the same notional principal maturing at times t = 1, 2,.., T and LN [ fFRA (0, t − 1) − i] zero-coupon bonds also maturing at times t = 1, 2,.., T.

Given that swaps are portfolios of FRAs and zeros, it makes sense that one can use swap rates to extract forward rates. This extraction is demonstrated in Extension 22.1.

The Yield and Swap Curve This section relates the Treasury (par bond) yield curve to the swap curve. Recall that the yield curve for US Treasury securities is obtained from T-bills for maturities of one year or less and from T-notes and T-bonds for maturities of more than one year. For the coupon-bearing bonds, the par bond yields are graphed. Recall that the par bond yields correspond to the yields on Treasury securities that are priced at par. Such a hypothetical Treasury yield curve is graphed in Figure 22.4.

SWAPS AND FRAS

553

EXTENSION 22.1: Computing Forward Rates from Swap Rates Computing forward rates from swap rates is analogous to computing forward rates from coupon bond prices (see Extension 21.3). This is done by solving a system of linear equations for a collection of unknowns. The first step is to compute the zero-coupon bond prices. Consider a par swap with value VSwap = 0, notional (LN ), maturity date (T), and a swap rate (iS ) paid every period. Setting expression (22.7) equal to (22.8) in Result 22.1 and rearranging terms gives us an expression involving the zero-coupon bond prices, which are the unknowns, that is, T

1 − B (0, T) =

[∑ i =1

B (0, t) × is ]

(1)

Next, suppose we are given a collection of T distinct maturity swap rates. Then, each swap has an equation similar to expression (1). This gives us T equations in the T unknowns {B(0, 1), B(0, 2),..., B(0, T)}. Solve them to obtain the zero-coupon bond prices. If the system does not have a solution, one can find those unknowns that minimize the sum of errors squared between the right side of expression (1) and the bond prices across all bonds.

EXT. 22.1 EX. 1: Computing Forward Rates from Swap Rates

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■

Today is time 0. Consider three plain vanilla fixed-for-floating swaps with swap rates is = 0.0309 for a swap maturing in one year, 0.0368 for a swap maturing in two years, and 0.0432 for a swap maturing in three years.

Computing Zero-Coupon Bond Prices ■

For the one-year swap, expression (1) gives 1 − B (0, 1) = B (0, 1) iS or B (0, 1) = 1/ (1.0309) = 0.97

■

For the two-year swap, expression (1) gives 1 − B (0, 2) = [B (0, 1) + B (0, 2)] iS or B (0, 2) = (1 − 0.97 × 0.0368) / (1 + 0.0368) = 0.9300

■

(2a)

(2b)

Finally, for the three-year swap, expression (1) gives 1 − B (0, 3) = [B (0, 1) + B (0, 2) + B (0, 3)] iS or B (0, 3) = (1 − 0.97 × 0.0432 − 0.9300 × 0.0432) / (1 + 0.0432) = 0.88

(2c)

These zero-coupon bond prices are the same as the corresponding values in Example 21.3, and we have already computed the forward rates in Example 21.5, completing this example.

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Now let’s construct a similar curve for the Eurodollar market. Recall that Eurodollar deposit rates are quoted for up to one year. These rates correspond to the T-bill yields and can be graphed directly. Par swap rates for swaps with maturities of more than one year complete the curve. Recall that a par swap rate is the rate that equates the present value of the cash flows from the swap’s underlying coupon bond (expression [22.8]) to the face value of the bond (the present value of the swap’s underlying floating-rate bond). Hence the par swap rate is equivalent to a par bond yield when viewed from this perspective. Such a hypothetical Eurodollar swap curve is also graphed in Figure 22.4. Thus, when constructed in this manner, the swap curve is almost identical to the Treasury yield curve. The difference is not in the formulas for the rates graphed—they are the same and provide an apples-to-apples comparison. Rather, the difference is due to the differing credit risks between the Treasury and Eurodollar rates as measured by the Treasury-Eurodollar (TED) spread. If it were plotted on the same graph, the Eurodollar rate would exceed the Treasury rate by the TED spread. In fact, one reason for graphing the two curves together is to visually see their differences (the TED spread)! When markets are calm, the TED spread tends to be small in magnitude and relatively constant across maturities. When markets are in turmoil, however, the TED spread becomes large in magnitude, with the biggest differences on the short end of the curve. It is a useful statistic to gauge the market’s pricing of credit risk for the largest European banks in times of calm and turmoil.

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FIGURE 22.4: Yield Curve and Swap Curve Interest rate Yield curve (Treasuries)

0

1 year Short-maturity bonds (yields)

Time Long-maturity bonds (par bond yields)

Interest rate Swap curve (Eurodollars)

0

1 year Short-maturity deposits (bbalibor rates)

Time Long-maturity swaps (par swap rates)

QUESTIONS AND PROBLEMS

22.7

Summary

1. The first swap was a currency swap arranged by Salomon Brothers between IBM and the World Bank. We divide the market developments into three categories: (1) the introduction of new swap contracts, (2) a shift from a brokerage business to a dealership market, (3) the founding of ISDA and the standardization of documents, and (4) the use of technology and the automation of trading. 2. There are three ways of initiating a swap contract: through phone calls, via a web-based platform, and by auctions. Auctions are more common in the case of municipal swaps. A swap position is closed through a buyout, through an assignment, or by entering into an offsetting swap. 3. A swap contract’s documents must describe the cash flows, a timetable for the cash payments, and other relevant considerations like how to terminate a swap early and what to do if one side defaults. 4. The value of a plain vanilla interest rate swap to the fixed-rate payer equals the present value of the floating-rate note minus the present value of the fixed-rate loan implicit in the swap. A par swap sets the fixed interest rate so that the swap has a zero value. 5. A swap can be synthetically constructed as a portfolio of forward rate agreements and zero-coupon bonds.

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6. The Treasury par bond yield curve and the Eurodollar swap curve graph identical rates, with the exception of credit risk. The difference between these two curves is called the Treasury over Eurodollar deposit rate (TED) spread.

22.8

Cases

Procter and Gamble versus Bankers Trust: Caveat emptor (Thunderbird Case

A06-05-0001, European Case Clearing House). The case analyzes a complex interest rate swap between Procter & Gamble and Bankers Trust. Wal-Mart’s Use of Interest Rate Swaps (Harvard Business School Case 108038-

PDF-ENG). The case studies Walmart’s use of interest rate swaps to hedge the fair value of its fixed-rate debt against changing interest rates. Banc One Corp.: Asset and Liability Management (Harvard

Business School Case 294079-PDF-ENG). The case examines a bank’s use of interest rate swaps to manage the sensitivity of its earnings to changes in interest rates and as an attractive investment alternative to conventional securities.

22.9

Questions and Problems

22.1. If you enter a plain vanilla interest rate swap, what risks do you face? 22.2. a. To reduce counterparty risk in a swap, what conditions are written into

the swap contract?

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b. Consider both the structure of the regular payments and the notional. Are

there any other restrictions in the ISDA swap documentation that reduce counterparty risk? 22.3. a. What are the three ways one can initiate a swap transaction? b. What are the four ways one can use to close a swap position? 22.4. Explain how a plain vanilla interest rate swap can change a fixed-rate loan into

a floating-rate loan, and include a diagram to support your explanations. 22.5. What is the value of a floating-rate loan? Explain your answer. 22.6. a. What is the warehousing of swaps? b. Have swap dealers been successful in managing their books of plain vanilla

interest swaps? Explain your answer. 22.7. Compute the value of a four-year fixed-rate loan with a coupon of 5 percent

paid yearly on a principal of $100. Use the zero-coupon bond prices in the following table:

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Times to Maturity T

Today’s Zero-Coupon Bond Prices B(0, T)

1

B(0, 1)

0.97

2

B(0, 2)

0.93

3

B(0, 3)

0.88

4

B(0, 4)

0.82

22.8. Explain why a cancellable swap might be of interest to a corporate treasurer. 22.9. Explain how to construct a swap using a floating-rate bond and a fixed-rate

bond. 22.10. Explain why a swap has zero value when it is initiated. Why might this be a

useful convention? 22.11. Based on the examples in the text, come up with an argument as to why plain

vanilla interest rate swaps do not exchange principal, while a plain vanilla currency swap does. 22.12. Consider the swap in Example 22.2. Now, assume that the payments are made

every six months. Zero-coupon bond prices (maturing every six months) are given in the following table:

QUESTIONS AND PROBLEMS

TABLE 22.3: Zero-Coupon Bond Prices Time to Maturity (in Years)

Zero-Coupon Bond Prices B (0, T)

0.5

B(0, 0.5)

0.99

1.0

B(0, 1.0)

0.97

1.5

B(0, 1.5)

0.95

2.0

B(0, 2.0)

0.93

2.5

B(0, 2.5)

0.91

3.0

B(0, 3.0)

0.88

Compute the value of this swap. 22.13. How can one synthetically construct a swap using a portfolio of forward rate

agreements and zero-coupon bonds? 22.14. Ignoring credit risk, is the swap rate on a T-year swap the same as the par-

bond yield on a T-year Treasury bond? Explain your answer. 22.15. Describe the two different ways of synthetically constructing a swap. Which

method is likely to be the easier to implement?

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22.16. In the Bankers Trust versus Procter & Gamble situation, P&G argued that

it did not fully understand the complex interest rate swap it had entered. Do you think it is prudent to enter a derivative that you do not completely understand? Explain your answer. 22.17. a In chapter 1, we listed five different kinds of risks that were considered

important by the Basel Committee on Banking Supervision and the International Organization of Securities Commissions. Pick three of these risks and explain why they are important to traders in the swap market. b Discuss the role played by the International Swaps and Derivatives Associ-

ation in the swap and over-the-counter derivatives markets. c What is a Master Agreement? 22.18. Can you explain why an amortized interest rate swap might be a useful

instrument for managing the risk of residential home mortgages? 22.19. After it is written and time has passed, will an interest rate swap always be

equal to zero value? Explain. 22.20. Why are the swaps considered excellent examples of financial engineering?

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23

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Single-Period Binomial Heath–Jarrow–Morton Model 23.1 Introduction

The Hedge Ratio (the Holy Grail)

23.2 The Basic Interest Rate Derivatives

Risk-Neutral Valuation Valuing a Floorlet Valuing Various Interest Rate Derivatives

Uses of Caps, Floors, and Collars

23.3 A Brief History of Interest Rate Derivatives Models 23.4 The Assumptions 23.5 The Single-Period Model

Multiple Factors

23.6 Summary 23.7 Appendix The Equal Pseudo-probability Condition

Arbitrage-Free Evolutions Zero-Coupon Bond Prices and Martingales Understanding the Equal Pseudo-probability Condition

Proof of Sufficiency Proof of Caplet and Floorlet Parity

23.8 Cases

Actual versus Pseudo-probabilities 23.9 Caplet Pricing

Questions and Problems

INTRODUCTION

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23.1

Introduction

This chapter introduces the Heath–Jarrow–Morton (HJM) model for valuing interest rate derivatives in the context of a simple binomial model. This chapter is similar to yet different from the binomial option pricing model in Chapter 17. It is similar in that the no-arbitrage and hedging arguments are invoked to value interest rate derivatives. It is different in that a key assumption is changed. No longer is there only one stock with random returns and one interest rate that is constant. Instead, we introduce a term structure of interest rates that fluctuates randomly across time. Just like Alice in her adventures in Wonderland,1 we have fallen into a new world, the world of interest rate derivatives. It is similar to but different from our more familiar world of equity derivatives. In this new world, like the Cheshire Cat and Mad Hatter, there are strange and unusual characters—forward rates and zero-coupon bond prices—that will help us in searching for our white rabbit, which is the interest rate derivative’s hedge ratio. Although we considered interest rate derivatives (forward rate agreements [FRAs] and swaps) in Chapters 21 and 22, they had linear payoffs. Consequently, we could value the derivatives without assuming a particular evolution for the term structure of interest rates. This and the next few chapters extend this analysis by considering the basic interest rate derivatives with nonlinear payoffs, including futures contracts as well as interest rate call and put options. In this new world, calls and puts are called caplets and floorlets. Portfolios of these options trade, called caps and floors. This chapter discusses caps and floors and their uses. A brief history of interest rate derivatives models follows. Then, we study the single-period binomial HJM model. Because the purpose of the HJM model is to price interest rate derivatives, just as in the binomial option pricing model of Chapter 17, we need to assume a stochastic evolution for the term structure of interest rates. The presentation of the material in this chapter purposely parallels the development of the binomial option pricing model (Chapter 17). As such, this chapter piggybacks on that material to make our presentation simpler and easier to follow. First we state the necessary assumptions. Second, to value a traded caplet, we construct a synthetic caplet with identical payoffs using a zero-coupon bond and a money market account (mma). The cost of construction yields the caplet’s arbitragefree price. This takes us to determining the hedge ratio, martingale pricing, and risk-neutral valuation. To value a floorlet, we introduce caplet and floorlet parity, the analogue of put–call parity. A discussion of the valuation of alternative interest rate derivatives concludes the chapter. Chapter 24 picks up where this chapter ends by extending the single period to a multiperiod setting, thereby generating an HJM model that is useful in practice. One last comment is necessary before reading Chapters 23 and 24. All computations in the next two chapters are performed using 16-digit accuracy. When reporting results in various formulas, however, we often round to four decimal places. The resulting equations, if evaluated using the reported (and rounded) numbers, may give different answers from those reported based on sixteen-plus digit accuracy. This 1

Alice’s Adventures in Wonderland is a classic book published in 1865 by Charles L. Dodgson under the pseudonym Lewis Carroll.

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is a typical problem when reporting numerical computations in this fashion (see Extension 2.1). Being forewarned, you should have no problems understanding the computations that follow.

23.2

The Basic Interest Rate Derivatives

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As seen earlier in this book, four basic derivatives are written on any underlying security or index: forwards, futures, call options, and put options. The same is true in the world of interest rate derivatives. In this context, the underlying “index” is the default-free spot rate of interest, and call and put options go by different names: caplets and floorlets, respectively. Yet in this strange new world, caplets and floorlets do not trade by themselves but only in portfolios called caps and floors, respectively. The reason these securities trade as portfolios is to economize on transaction costs. These securities, as will be shown, are useful for hedging fixed- and floating-rate coupon bonds. As coupon bonds have multiple cash flows, it makes sense that the basic hedging instruments need multiple cash flows as well. Interestingly, in this interest rate world, forwards trade separately as FRAs and also in portfolios, called interest rate swaps. We summarize these four basic derivatives in Table 23.1 and describe them subsequently. ■

Forwards. A forward rate agreement (FRA) is an over-the-counter (OTC) contract that fixes an interest rate for borrowing (or lending) over some future time period. The FRA rate is equivalent to the forward rate, which is the fundamental building block for our theory. An interest rate swap is a portfolio of zero-coupon bonds and FRAs (see Chapter 22).

■

Futures. An interest rate futures is an exchange-traded contract that fixes a borrowing (or a lending) rate at some future date. It differs from an FRA the same way that a futures contract differs from a forward contract. A host of features such as standardization, a clearinghouse, margins, and daily settlements make futures safe and easy to trade. Some interest rate futures (like those on Treasuries) end in physical delivery, while others (like those on Eurodollars) end in cash settlement.

■

Call options. A caplet is a European call option on an interest rate that is effective for a single period. The buyer gets paid the spot rate minus the cap rate when it’s positive, and nothing otherwise. A cap is a portfolio of caplets. Caps are cash settled contracts usually based on a libor rate index.

■

Put options. A floorlet is a European put option on an interest rate that is effective for a single period. The buyer is paid that period’s floor rate minus the spot rate only when it’s positive. A floor is a portfolio of floorlets. Floors are cash settled contracts usually based on a libor rate index.

Of course, other, more exotic interest rate derivatives trade. One of these is a swaption, which is an option with an interest rate swap as the underlying. They come in two basic types: a payer swaption is an option to enter into a swap where the holder pays the fixed rate and receives the floating rate, while a receiver swaption is an option to enter into a swap where the holder pays the floating rate but receives the fixed rate.

THE BASIC INTEREST RATE DERIVATIVES

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TABLE 23.1: The Basic Interest Rate Derivatives Today

Intermediate Date

Maturity Date

1. Forwards: forward rate agreements (FRAs) Buyer negotiates FRA rate (which is equivalent to the forward rate) and contract terms with the seller. No cash changes hands today.

No cash flows.

Short pays buyer the spot rate minus the FRA rate, applied on a notional amount and computed over a fixed time interval.

2. Futures: Eurodollar futures contracts Regulated, standardized contracts betting on a simple interest rate (libor) to be realized at a future date. No cash changes hands today.

Marking-to-market cash flows.

Cash settlement based on a notional quantity computed over a fixed time period.

3. Call options: caplets A cap buyer pays a premium and buys a portfolio of European calls of different maturities on the same notional principal. The underlying is a simple interest rate (libor). A caplet is one of those European calls.

Caplet buyer receives the spot rate less the cap rate if positive, or zero otherwise.

Same as in intermediate date.

4. Put options: floorlets A floor buyer pays a premium and buys a portfolio of European put options of different maturities on the same notional principal. The underlying is a simple interest rate (libor). A floorlet is one of those European puts.

Floorlet buyer receives the floor rate minus the spot rate if positive, or zero otherwise.

Same as in intermediate date.

Although the pricing of caps and floors is the primary goal in this chapter and the next two, we also discuss how to price swaptions. Before discussing pricing, let us consider how financial institutions use caps and floors.

Uses of Caps, Floors, and Collars Example 23.1 illustrates how an interest rate cap can be used to speculate or hedge interest rate risk.

EXAMPLE 23.1: An Interest Rate Cap

Speculation ■

International Treats and Restaurants Inc.’s (INTRest, a fictitious name) treasurer buys a cap today (time 0) with a cap rate k = 5 percent per year on a notional of LN = $100 million. This OTC contract matures in two years and has the six-month libor rate index as the underlying floating rate of interest. This cap is a collection of four caplets, each of which makes payments based on the realized spot rates at six-month intervals.

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Figure 23.1 shows the cap’s payoffs. The dotted line shows the six-month libor rate index. Spot rates are determined at the beginning of each time period, and payments based on these rates are made at the end of each time period. - Today’s spot rate R(0) is 0.06. The first caplet pays INTRest after six months: LN × [R (0) − k] × Time period = 100 × (0.06 − 0.05) × 0.5 year = $0.5 million

(23.1a)

- Six months later (time 1), the six-month libor rate index is R(1) = 0.05. The second caplet is at-the-money and no payment is made. - As R(2) = 0.04 is less than the cap rate, the third caplet also expires worthless at time 3. - The spot rate R(3) is 0.07 after a year and a half. The fourth caplet gives INTRest a payoff of 100 × (0.07 – 0.05) × 0.5 year = $1 million at time 4.

FIGURE 23.1: A Cap Buyer’s Payoffs

[R(0) – k] = 0.01 determined at time 0 but paid at time 1

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Cap payoff Spot rate

0.07 0.06 k = 0.05

Cap Rate

0.04

0 Today

[R(3) – k] = 0.02 determined at time 3 but paid at time 4 1 2 6 months

3

4 Time t 2 years

Hedging A cap protects the holder against interest rate increases by putting a ceiling on the floating borrowing rate. To see this, suppose INTRest buys this cap and simultaneously issues floating rate debt paying the libor rate index plus 50 basis points with L = $100 million par value and with a similar payment date structure as the cap.

■

THE BASIC INTEREST RATE DERIVATIVES

- After six months, INTRest’s cash flow is Cash flow from bond + Cash flow from first caplet = −L × [R (0) + 0.005] × Time period + LN × [R (0) − k] × Time period = −100 × (0.005 + k) × 0.5 = 2.75 million

(23.1b)

- The second and third cash out flows are just $2.75 million and $2.25 million, respectively, because INTRest benefits from the low floating rates at times 1 and 2, with R(1) = 0.05 and R(2) = 0.04. - At time 4, the total payment is again capped at $2.75 million. ■

The cap places a ceiling on INTRest’s borrowing cost at the cap rate + 50 basis points.

The cap’s payoff given in Equation 23.1a can be formalized as follows. Consider a cap dates t = 1, 2,..., T. If the cap has a notional of LN and a cap rate k per year, then a caplet maturing at time t has a payoff equal to

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ct = max [0, R (t − 1) − k] × LN × Time period

(23.2a)

where R(t – 1) is the spot rate (libor or euribor in most cases) determined at time (t – 1) that applies to the time period [t – 1, t]. This time period is determined according to market conventions (such as actual/360 for Eurodollars and Euribor or actual/365 for pound sterling deposits). The cap’s payoff is the sum of the individual caplet’s payoffs. In our discrete time model, the tth caplet’s payoff at time t simplifies to ct = max [0, R (t − 1) − k]

(23.2b)

where we have assumed that the notional principal is $1 and the time period has unit length. A floor allows the holder to speculate or hedge on an interest rate decrease. A floor is a collection of floorlets, which are European put options on interest rates. A floor can ensure a minimum interest rate on an existing bond holding (which is analogous to a protective put strategy from Chapter 15) or if one simultaneously buys the bond and a floor (which is analogous to a married put). Many endowment funds, hedge funds, mutual funds, pension funds, and financial institutions buy OTC traded interest rate floors to achieve a minimum return on their bond holdings while retaining the upside potential. We can similarly describe a floorlet’s payoff. Consider a floor consisting of T floorlets with maturity dates t = 1, 2, . . . , T. If the floorlets have a notional of LN and a floor rate k per year, then a floorlet maturing at time t has a payoff equal to pt = max [0, k − R (t − 1)] × LN × Time period

(23.3a)

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where R(t – 1) is the spot rate determined at time (t – 1) that applies to the time period [t – 1, t]. Similarly, a floor’s payoff is the sum of the individual floorlet’s payoffs. Analogously, a floorlet’s payoff at time t in the case of our discrete model with a notional principal of $1 and a time period of unit length is pt = max [0, k − R (t − 1)]

(23.3b)

A commonly used combination of interest rate derivatives is an interest rate collar that combines a cap and a floor with identical terms and conditions but with different strike rates. Let’s consider Example 23.2.

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EXAMPLE 23.2: An Interest Rate Collar ■

Suppose that INTRest Inc.’s treasurer buys the interest rate cap of the previous example. It consists of four caplets each with a notional LN of $100 million. The cap rate k1 is 5 percent per year, the maturity date is 2 years, the six-month libor rate index is the underlying floating rate of interest, and the payments are based on the realized spot rates at six-month intervals.

■

Next, the treasurer thinks that she can save the company money by selling a floor with identical terms, except a floor rate of k2 = 2 percent per year. The premium earned on the floor offsets the cap’s cost and lowers the effective cost of the overall position.

■

Now if the realized spot rate exceeds 5 percent, INTRest will receive a payment from the cap so that the company’s effective rate is 5 percent. Conversely, if the spot rate falls below 2 percent, say, to 0.5 percent, INTRest will have to pay on the floor – (k2 – R) = – 0.015 or an outflow of 1.50 percent. The company has confined the interest rate paid to within the band of k2 to k1 or 2 to 5 percent. The short floor traded surrenders the opportunity to benefit from very low interest rates.

■

This combination of caps and floors is similar to collars in the equity market introduced in Chapter 15, and if the prices of the caps and floors match, then it is a zero-cost collar.

Before pricing interest rate derivatives like caps and floors, let us begin by providing a bird’s-eye view of the historical development of interest rate derivatives pricing models.

23.3

A Brief History of Interest Rate Derivatives Models

In their book Financial Calculus—An Introduction to Derivatives Pricing, Baxter and Rennie (1996, p. 203) wrote, In the interest-rate setting, Heath–Jarrow–Morton is as seminal as Black–Scholes. By focusing on forward rates and especially by giving a careful stochastic treatment, they produced the most general (finite) Brownian interest-rate model possible. Other models may claim differently, but they are just HJM with different notations.

THE ASSUMPTIONS

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To understand the relevance of these comments, we step back and look at the history of interest rate derivatives models.2 Interest rate derivatives pricing models began with a collection of papers focusing on the evolution of the spot rate of interest, called “spot rate” models, and it eventually ended with the HJM model. ■

Spot rate models. This class of models focused on the evolution of the spot rate, R(t). These models include those by Vasicek (1977), Brennan and Schwartz (1979), and Cox et al. (1985). Spot rate models have two major limitations. First, in these models, the values of interest rate derivatives depend on interest rate risk premia or, equivalently, on the expected return of default-free bonds. Recall that estimating a risk premium was the nagging problem facing option pricing in the pre–Black–Scholes–Merton (BSM) era, which the BSM model overcame. Second, these models can not easily match the initial yield curve. This yield curve matching is essential for accurate pricing and hedging of interest rate derivatives because any discrepancies in yield curve matching may generate “false” arbitrage opportunities in the priced derivatives.

■

HJM models. To address the limitations of the spot rate model, the HJM model was developed. Motivated by consulting, Professor David Heath of Cornell University’s School of Operations Research and Industrial Engineering and a PhD student, Andrew Morton, were working on this problem. Robert Jarrow, a professor of finance at Cornell’s Johnson Graduate School of Management, joined them. Together they developed the HJM model, which was first presented as a working paper in 1987 and published as a series of papers in 1990 and 1992. The HJM model is a very general continuous-time and multifactor model. In fact, all of the previous spot rate models are special cases. A popular subcase of the HJM model known as the libor model (or the market model or the BGM model) is widely used by Wall Street for the pricing and hedging of caps, floors, and swaptions. Sandmann et al. (1995), Miltersen et al. (1997), and Brace et al. (1997) independently developed the libor model. This model generates a formula for a caplet that is analogous to the BSM European call option formula. As such, it is familiar territory for derivatives traders, which explains its widespread popularity (see Chapter 25 for a discussion of this model).

23.4

The Assumptions

The assumptions underlying the single binomial HJM model are similar to those used to generate the binomial option pricing model of Chapter 17, but with an important modification. The old assumption A6 that there is no interest rate uncertainty is now relaxed, and the old assumption A7 is replaced with a new assumption characterizing the random evolution of the term structure of interest rates. This new assumption is similar to, but more complex than, the old assumption A7 describing the evolution of stock prices.

2

See Hughston (1996) and Ho and Lee (2004), Chapter 5, for additional histories of interest rate derivatives models.

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For completeness, we first restate assumptions A1–A5. A1. No market frictions. We assume that trading involves no market frictions such as transaction costs (brokerage commissions and bid/ask spreads), margin requirements, short sale restrictions, and taxes (which may be levied on different securities at different rates). Moreover, assets are perfectly divisible. This helps us to create a benchmark model to which one can later add frictions, and it’s a reasonable approximation for institutional traders. A2. No credit risk. We assume that there is no credit risk. Also known as default risk or counterparty risk, this is the risk that the counterparty will fail to perform on an obligation. The absence of credit risk is a reasonable assumption for both exchangetraded derivatives and many OTC derivatives because of collateral provisions. A3. Competitive and well-functioning markets. In a competitive market, traders act as price takers and prices cannot be manipulated. In a well-functioning market, price bubbles (which create a wedge between an asset’s market price and its intrinsic or fundamental value) are assumed to be absent. A4. No intermediate cash flows. Many assets reward their holders with cash flows such as dividends for stocks and coupon income for bonds. For now we assume that assets provide no such cash flows to their holders. As you will see later, such cash flows are easily incorporated into the model.

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A5. No arbitrage opportunities. This is self-explanatory. We now relax the old assumption A6 (of Chapter 17) that there is no interest rate uncertainty. Instead, we allow the entire term structure of interest rates to fluctuate randomly through time. The relevant term structure is that of forward rates. Recall that Equation 21.9 in Chapter 21 gives the definition of the time t forward rate for the future period [T – 1, T] as f (t, T − 1) =

B (t, T − 1) −1 B (t, T)

(23.4)

where B(t, T) is the time t price of a zero-coupon bond maturing at time T. When the time t forward rate corresponds to the immediate future period [t, t + 1], the forward rate gets a special name, the spot rate, and is denoted by f (t, t) = R(t). Given the term structure of forward rates, one can easily compute the term structure of zero-coupon bond prices as shown in Chapter 21 (see Result 21.2). Conversely, given the term structure of zero-coupon bond prices, one can easily obtain the term structure of forward rates using Equation 23.4. This equivalence implies that the term structure of forward rates and the term structure of zero-coupon bond prices contain the same information. Thus one can describe the evolution of the term structure of interest rates by imposing an assumption on either the forward rates or the zero-coupon bond prices. In discrete time models, for pedagogical reasons—to see the economic reasoning more clearly—it is easier to impose the term structure evolution assumption on the zero-coupon bond prices. For parameter estimation, however, one must transform

THE ASSUMPTIONS

the zero-coupon bond price evolution to that of the forward rates because the forward rates’ evolution is more time stationary, a property compatible with statistical estimation procedures (see Chapter 21). In continuous-time models, however, the situation reverses. It is easier to impose the term structure evolution assumption directly on forward rates. This is the approach taken in Chapter 25 when discussing the HJM libor model. Of course, as we now know, these are equivalent ways of imposing the same assumption. To model the term structure evolution, two changes are made to the binomial tree of Chapters 17 and 18. First, the spot rate of interest is now random across time; call it R(t). Second, stock prices are irrelevant and dropped from the model. Instead, each node now has a vector of zero-coupon bond prices B(t, T) with maturities T = 1, 2, 3, . . . for times t = 0, 1, 2, . . . , T

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A6. Single-period binomial process. The zero-coupon bond prices B(t, T) for t = 0, 1 and T = 1, 2, 3 trade in discrete time, and their prices evolve according to a binomial process. After time 0, there are two states called “up” and “down.” First, if the zero-coupon bond matures at time 1, then the current bond price goes to $1 regardless of the state. Second, the T > 1 maturity zero-coupon bond price is multiplied by the factor u(0, T) in the up state and takes the price u(0, T) × B(0,T) with the actual probability q(0) > 0, or it is multiplied by the factor d(0, T) in the down state and takes the price d(0, T) × B(0, T) with probability 1 – q(0). We assume that u(0, T) > d(0, T). The single-period evolution for the term structure of zero-coupon bond prices is given in Figure 23.2. We first explain this evolution via an example (see Example 23.3).

EXAMPLE 23.3: The Single-Period Term Structure Evolution Let today be time 0. There are four securities in the model: three zero-coupon bonds and a moneymarket-account (mma). We describe their evolution over a one-year period.

The Zero-Coupon Bond Prices ■

The time 0 prices of zero-coupon bonds maturing in 1 year (time 1), 2 years (time 2), and 3 years (time 3), respectively, are given in the following matrix: ⎡B (0, 3) = $0.88⎤ ⎢B (0, 2) = $0.93⎥ ⎢ ⎥ ⎣B (0, 1) = $0.97⎦

(23.5)

These prices are taken from the common zero-coupon bond price data used in Example 21.1. To reduce clutter, we drop the dollar signs from the tree.

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■

After one period, bond prices can move up or down. We assume that this happens with the actual probabilities q(0) = 0.3 and [1 – q(0)] = 0.7, respectively. The bond prices in the up state (labeled by the subscript u) are ⎡B (1, 3)u = 0.9167⎤ ⎢B (1, 2) = 0.9620⎥ ⎢ ⎥ u ⎣B (1, 1)u = 1 ⎦

(23.6a)

The bond prices in the down state (labeled by the subscript d) are ⎡B(1, 3)d = 0.8977⎤ ⎢B(1, 2) = 0.9555⎥ d ⎢ ⎥ ⎣B(1, 1)d = 1 ⎦

(23.6b)

Note that at time 1, the bond maturing at time 1 pays $1 in both the up and down states and then disappears from the market.

Risky Versus Riskless Securities

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■

In this evolution, there are two riskless securities, the bond that matures at time 1 and the mma. Both the two- and three-period maturity bonds are risky. This can be seen by computing their returns in both the up and down states and noticing that they are different: ⎡[B(1, 3)u − B (0, 3)] /B (0, 3) = (0.9167 − 0.88) /0.88 = 0.0418⎤ ⎢ ⎥ ⎢[B(1, 2)u − B (0, 2)] /B (0, 2) = (0.9620 − 0.93) /0.93 = 0.0344⎥ ⎣[B(1, 1)u − B (0, 1)] /B (0, 1) = (1 − 0.97) /0.97 = 0.0309 ⎦

(23.7a)

⎡[B(1, 3)d − B (0, 3)] /B (0, 3) = (0.8977 − 0.88) /0.88 = 0.0201⎤ ⎢ ⎥ ⎢[B(1, 2)d − B (0, 2)] /B (0, 2) = (0.9555 − 0.93) /0.93 = 0.0274⎥ ⎣[B(1, 1)d − B (0, 1)] /B (0, 1) = (1 − 0.97) /0.97 = 0.0309 ⎦

(23.7b)

This observation will prove useful later in this section.

The lower part of Figure 23.2 gives the formalization of this example using the symbols introduced in assumption A6.

THE ASSUMPTIONS

FIGURE 23.2: One Period Evolution of Bond Prices in A Binomial Framework Time 0

Time 1

Zero-Coupon Bond Prices (Example) q(0) = 0.3 B(0, 3) = 0.88 B(0, 2) = 0.93 B(0, 1) = 0.97

B(1, 3)u = 0.9167 B(1, 2)u = 0.9620 B(1, 1)u = 1

(“Up” state)

B(1, 3)d = 0.8977 B(1, 2)d = 0.9555 B(1, 1)d = 1

(“Down” state)

R(0) = 0.0309

Zero-Coupon Bond Prices (Symbols)

q(0) B(0, 3) B(0, 2) B(0, 1)

u(0, 3)B(0, 3) = B(1, 3)u u(0, 2)B(0, 2) = B(1, 2)u 1 = B(1, 1)u

R(0)

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d(0, 3)B(0, 3) = B(1, 3)d d(0, 2)B(0, 2) = B(1, 2)d 1 = B(1, 1)d Money Market Account (mma) A(0) = 1

A(1) = [1 + R(0)] = 1.0309

THE ZERO-COUPON BOND PRICE EVOLUTION ■

Today is denoted by time 0. The zero-coupon bond prices are given by B(0, 1), B(0, 2), and B(0, 3). They pay a dollar on their maturity dates: times 1, 2, and 3, respectively. Only time 1 is shown in this evolution.

■

The actual probability of moving “up” is q(0) > 0.

■

At time 1, the prices of the zeros move to the up state (in which case they get multiplied by the up factor) or the down state (where they get multiplied by the down factor). - If zero-coupon bond prices go up, the three-period bond has a new price of B(1, 3)u , which is obtained by multiplying B(0, 3) by the up factor u(0, 3). - Similarly, the two-period bond’s price is u(0, 2)B(0, 2) = B(1, 2)u .

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- The one-period bond’s price is B(1,1)u = 1 because it matures, pays $1, and disappears from the market. - Note that the subscript u is attached to the prices of the zeros at time 1 to distinguish them from the down state prices. - If zero-coupon bond prices go down, the three-period bond attains a new price of d(0, 3)B(0, 3) = B(1, 3)d . Similarly, d(0, 2)B(0, 2) = B(1, 2)d and B(1, 1)d = 1. Here we use the subscript d for the down state.

THE SPOT RATE ■

Irrespective of the state that occurs, the shortest-maturity bond pays $1 at time 1. This bond is thus riskless and earns the spot rate: R (0) = [1 − B (0, 1)] /B (0, 1) = (1 − 0.97) /0.97 = 0.0309 or 3.09 percent

(23.8a)

THE MONEY MARKET ACCOUNT By definition, the mma grows at each period’s spot rate of interest. If one invests $1 in the mma at time 0, it pays a dollar return of [1 + R(0)] at time 1. Hence the mma’s values at times 0 and 1 are A (0) = $1 and A (1) = 1 [1 + R (0)] = $1.0309

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23.5

(23.8b)

The Single-Period Model

We now present the single-period binomial HJM model. We apply this model to price caplets, although as discussed later, the model is easily used to price other interest rate derivatives as well. The first step in this methodology is to guarantee that the assumed evolution is arbitrage-free.

Arbitrage-Free Evolutions “The rain came down, the streams rose, and the winds blew and beat against that house; yet it did not fall, because it had its foundation on the rock,” but the house built on sand “fell with a great crash” (see Matthew 7:25 and 7:27, King James New International Version). Similarly, we want to build a robust binomial tree that will not crumble when used to price interest rate derivatives. The tree (the method) crumbles if the tree’s evolution admits arbitrage. Why? Because how can one determine the arbitrage-free price for a derivative when there exist arbitrage opportunities in the prices of the zero-coupon bonds themselves? Obviously, one cannot. Hence we need to guarantee that the assumed evolution is “built on rock” by being arbitrage-free. The task is far more complex than it was in the binomial model for pricing equity derivatives because here we have to ensure that a trader cannot make arbitrage profits by trading any combination of these zero-coupon bonds.

NECESSARY CONDITIONS We can use our insights obtained from the binomial model in Chapter 17 to determine the arbitrage-free conditions. In

THE SINGLE-PERIOD MODEL

Chapter 17, there was only one risky asset’s price on the binomial tree. Recall the no-arbitrage condition using the notation of this chapter: Up factor > 1 + R (0) > Down factor Here, however, we have two risky bonds at each node. Hence, if there is no arbitrage, then this condition must be true for both of the risky zero-coupon bonds B(0,2) and B(0,3). Otherwise, just as argued in Chapter 17, one can form a portfolio of a risky zero-coupon bond and the mma to form an arbitrage opportunity. This logic implies then that a necessary condition for no arbitrage is u (0, T) > 1 + R (0) > d (0, T)

for T = 2, 3

(23.9a)

where u(0, T) and d(0, T) are the T-period zero-coupon bond’s up and down factors from time 0 to time 1. But this condition is not sufficient to guarantee that the tree is arbitrage-free. The reason is because it only relates each risky bond to the mma. It does not compare the two risky bonds’ returns to each other. Such a comparison will generate another necessary condition, which, in conjunction with Equation 23.9a, will turn out to be sufficient. Before determining that remaining condition, however, it is useful to rewrite Equation 23.9a in an alternative form.

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ALTERNATIVE NECESSARY CONDITIONS Simple logic shows that Equation 23.9a is true if and only if there exists a real number 𝜋(0,T) with 0 < 𝜋(0, T) < 1 such that 1 + R (0) = 𝜋 (0, T) u (0, T) + [1 − 𝜋 (0, T)] d (0, T)

(23.9b)

for T = 2, 3. That is, the dollar return on the mma is a weighted average of the up and down returns on the risky zero-coupon bonds. Note that the weight 𝜋(0, T) depends on the current time (which is given by the first argument 0) and the particular risky bond under consideration (which is given by the second argument T). Solving Equation 23.9b gives a formula for this weight: u (0, T) − [1 + R (0)] 1 + R (0) − d (0, T) and 1 − 𝜋 (0, T) = ( u (0, T) − d (0, T) ) ( u (0, T) − d (0, T) ) (23.9c) for T = 2, 3. With this formula, the necessary condition for no arbitrage simplifies to just checking to see if 0 < 𝜋(0, T) < 1 for each T. Last, we can rewrite Equation 23.9b one more time. Dividing both sides of (23.9b) by [1 + R(0)] and multiplying both sides by B(0, T) gives our final result: 𝜋 (0, T) =

𝜋 (0, T) B (0, T) u (0, T) + [1 − 𝜋 (0, T)] B (0, T) d (0, T) for T = 2, 3 1 + R (0) (23.9d) We will return to this condition shortly and explain its significance. B (0, T) =

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SUFFICIENT CONDITIONS The previous necessary conditions (23.9a)–(29.9d) do not compare the dollar returns of the two risky bonds. To compare these dollar returns, to see if there is any arbitrage embedded in their price evolutions, we can compare the cash flows to the traded two-period zero with a synthetic two-period zero, constructed using a portfolio of the three-period zero and the mma. If the price of the traded two-period zero equals the cost of constructing the synthetic two-period zero, then we know the dollar returns on the two- and three-period zeros admit no arbitrage. It is shown in the appendix to this chapter that for this to be true, B (0, 2) =

𝜋 (0, 3) B (0, 2) u (0, 2) + [1 − 𝜋 (0, 3)] B (0, 2) d (2) 1 + R (0)

(23.10)

Equation 23.9d for the two-period zero is identical to Equation 23.10 but with 𝜋(0,2) instead of 𝜋(0,3). This can only happen if 𝜋 (0, 3) = 𝜋 (0, 2) with 0 < 𝜋 (0, 2) < 1

(23.11)

which is our last necessary condition. It turns out that Equation 23.11 is also a sufficient condition for the tree to be arbitrage-free. The proof of this assertion is contained in the appendix to this chapter. We summarize this discussion as a result (see Result 23.1).

RESULT 23.1 Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

Equal Pseudo-probability Condition The single-period zero-coupon bond price evolution is arbitrage-free if and only if there exists a real number 𝜋(0) strictly between 0 and 1 such that 𝜋 (0) = 𝜋 (0, T) for T = 2, 3

(23.12)

where 𝜋(0, T) = ([1 + R(0) – d(0, T)]/[u(0, T) – d(0, T)]), R(0) is the time 0 spot rate, and [u(0, T), d(0, T)] are the dollar returns on the T maturity zero-coupon bonds in the up and down states, respectively.

Because 𝜋(0) is a number between zero and one, we can interpret it as a probability, and we call it the pseudo-probability. We call Equation 23.12 the equal pseudoprobability condition. Fortunately, it greatly simplifies the task of checking to see if a zero-coupon bond price tree is arbitrage-free.

THE SINGLE-PERIOD MODEL

EXAMPLE 23.4: Verifying That a Single-Period Binomial Tree Is Arbitrage-Free Consider the term structure evolution given in Example 23.2 and Figure 23.2. To check whether this evolution is arbitrage-free, we compute the up and down factors for the two zero-coupon bonds as they evolve from time 0 to time 1. ■

For the zero-coupon bond maturing at time 3, we have up factor, u (0, 3) = B (1, 3)u /B (0, 3) = 0.9167/0.88 = 1.0418 and down factor, d (0, 3) = B (1, 3)d /B (0, 3) = 0.8977/0.88 = 1.0201

■

For the zero-coupon bond maturing at time 2, we have up factor, u (0, 2) = B (1, 2)u /B (0, 2) = 0.9620/0.93 = 1.0344 and down factor, d (0, 2) = B (1, 2)d /B (0, 2) = 0.9555/0.93 = 1.0275 Next, we compute the pseudo-probabilities for the two- and three-period zeros: 1 + R (0) − d (0, 3) 1.0309 − 1.0201 = = 0.5 ( u (0, 3) − d (0, 3) ) (1.0418 − 1.0201) 1 + R (0) − d (0, 2) 1.0309 − 1.0275 𝜋 (0, 2) = = = 0.5 ( u (0, 2) − d (0, 2) ) (1.0344 − 1.0275)

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𝜋 (0, 3) =

■

Because both these pseudo-probabilities are equal and strictly between 0 and 1, the single-period tree is arbitrage-free.

■

For this tree, 𝜋(0) = 0.5 is the unique pseudo-probability of an up movement for zero-coupon bonds maturing at times 2 and 3.

Zero-Coupon Bond Prices and Martingales Using the new notation, under the no-arbitrage condition (23.12), we can rewrite Equation 23.9d as B (0, T) =

𝜋 (0) B(1, T)u + [1 − 𝜋 (0)] B(1, T)d for T = 2, 3 1 + R (0)

(23.13a)

Suppose we view the “probability” of B(0, T) going up as 𝜋(0) and the probability of going down as [1 – 𝜋(0)], where T = 2 and 3. Then, this equation shows that the zero-coupon bond’s expected discounted payoff using the pseudo-probabilities is today’s zero-coupon bond price! This gives a method for computing the present value of a zero-coupon bond. Notice how similar this is to Equation 17.5 in Chapter 17. There the stock’s price evolved as a martingale so that the stock price equaled its discounted expected

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payoff at time 1 using the pseudo-probabilities. Now we do the same for both zerocoupon bonds. The final step is applying the pseudo-probabilities to price interest rate derivatives. It is hoped that you are beginning to understand why martingale pricing lies at the heart of modern derivatives valuation.

EXAMPLE 23.5: Bond Prices as Expected Discounted Values ■

Consider the term structure evolution given in Example 23.3 and Figure 23.2. At time 1, the threeperiod zero’s price is B(1,3)u in the up state with pseudo-probability 𝜋(0) and B(1,3)d in the down state with pseudo-probability [1 – 𝜋(0)]. As a dollar invested in the mma today gives [1 + R(0)] at time 1, application of Equation 23.13a gives 𝜋 (0) B(1, 3)u + [1 − 𝜋 (0)] B(1, 3)d 0.5 × 0.9167 + 0.5 × 0.8977 = 0.88 = 1.0309 1 + R (0) which matches B(0,3), today’s price of the zero-coupon bond maturing at time 3.

■

Applying the same formula to the two-period zero whose prices at time 1 are B(1, 2)u and B(1, 2)d with pseudo-probabilities 𝜋(0) and [1 – 𝜋(0)], respectively, we get 𝜋 (0) B(1, 2)u + [1 − 𝜋 (0)] B(1, 2)d 0.5 × 0.9620 + 0.5 × 0.9555 = = 0.93 1.0309 1 + R (0)

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which matches B(0,2), today’s price of the zero-coupon bond maturing at time 2.

Using now familiar abstract notation, we rewrite these equations as the following result (see Result 23.2).

RESULT 23.2 Risk-Neutral Valuation for Zero-Coupon Bond Prices The time 0 value of a zero-coupon bond is B (0, T) =

E𝜋 [B (1, T)] for T = 2, 3 1 + R (0)

(23.13b)

where E𝜋 [.] denotes expectation using the pseudo-probability 𝜋(0), R(0) is the time 0 spot rate, and B(1, T) is the T maturity zero-coupon bond’s price at time 1.

This result shows that the time 0 price of a zero-coupon bond equals its expected discounted time 1 price, where the discount rate is the spot rate and the expectation is computed using the pseudo-probabilities.

THE SINGLE-PERIOD MODEL

Finally, using the notation for the value of the mma also gives B (1, T) B (0, T) for T = 2, 3 = E𝜋 ( A (0) A (1) )

(23.13c)

This equation shows that the zero-coupon bond’s price, normalized by the value of the mma, is a martingale under the pseudo-probabilities. Alternatively stated, a necessary and sufficient condition for the tree to be arbitrage-free is the existence of a real number 𝜋(0) strictly between 0 and 1 such that the normalized zero-coupon bond prices are martingales under this pseudo-probability. It is this form of the no-arbitrage condition that readily generalizes to more complex models for the evolution of the term structure of interest rates.

Understanding the Equal Pseudo-probability Condition The equal pseudo-probability condition has a nice economic interpretation. To understand this interpretation, we note the obvious fact that 𝜋 (0, 3) = 𝜋 (0, 2) holds if and only if [1 − 𝜋 (0, 3)] = [1–𝜋 (0, 2)]

(23.14a)

Writing this equation out in full using Result 23.1, we see that the tree is arbitragefree if and only if

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u (0, 3) − [1 + R (0)] u (0, 2) − [1 + R (0)] = ( u (0, 3) − d (0, 3) ) ( u (0, 2) − d (0, 2) )

(23.14b)

First, note that the numerator in Equation 23.14b is the compensation for bearing the risk of a zero-coupon bond versus holding the mma. This is because the possibility of receiving the larger return on the risky zero in the up state u(0, T) is the reason one holds a risky bond instead of the mma. Figure 23.3 illustrates this difference.

FIGURE 23.3: The Bond’s Risk Premium u(0,T)

reward $1

R(0)

d(0,T) risk

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Second, the denominator measures the spread, or the risk, of a zero-coupon bond. Indeed, the larger is the distance between the up and down returns on the zerocoupon bond, the riskier is the payoff, because both the gains and losses are magnified. Hence the distance between the largest and worst dollar returns on a zero-coupon bond is a measure of risk. Combined, the ratio measures the reward per unit of risk, called the risk premium. The no-arbitrage condition can thus alternatively be stated as follows: the tree is arbitrage-free if and only if all zero-coupon bonds earn the same risk premium. This last observation makes good economic sense. If the zero-coupon bonds had different risk premiums, traders would desire to hold the zero with the largest compensation per unit of risk. In fact, if a trader also cleverly shorts a bond with a smaller compensation per unit of risk, all of the interest rate risk can be removed from the resulting portfolio to generate an arbitrage opportunity. This is the logic underlying Result 23.2.

Actual versus Pseudo-probabilities As in Chapter 17, the pseudo-probabilities 𝜋(0) and 1 – 𝜋(0) for the up and down movements in the tree are both strictly positive if and only if the actual probabilities q(0) and 1 – q(0) are strictly positive. This condition implies that there exist two strictly positive numbers 𝜙u and 𝜙d such that 𝜋 (0) = 𝜙u q (0) and

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1 − 𝜋 (0) = 𝜙d [1 − q (0)] These numbers 𝜙u and 𝜙d transform the actual probabilities to the pseudoprobabilities, which are useful for pricing derivatives using the technique of riskneutral valuation (see below). The quantities 𝜙u and 𝜙d adjust the actual probabilities for the risk of the cash flows when computing the arbitrage-free price as a discounted expected value. The proof of this statement is identical to the proof given for the binomial model of Chapter 17 (see the appendix to that chapter) and is therefore omitted.

Caplet Pricing Having built this structure, it is now easy to price a caplet using the single-period HJM model. A caplet is a European call option on the spot rate of interest. Thus a caplet’s payoff with maturity date 2 and strike rate k is given by c (2) = max [R (1) − k, 0]

(23.15a)

Note that the caplet’s payoff at time 2 is based on the spot rate at time 1. This is by market convention.

THE SINGLE-PERIOD MODEL

Because this payoff is known at time 1, the present value of this payoff at time 1 is obtained by discounting (23.15a) by the spot rate R(1): c (1) =

max [R (1) − k, 0] 1 + R (1)

(23.15b)

Thus, we can alternatively view the caplet as having the maturity date 1 with this revised payoff. This is justified because all of the payoff ’s uncertainty is resolved at date 1. As in this example, to facilitate computation, we introduce the following convention in this and the next chapter. Caplet and Floorlet Maturity Convention: For valuation purposes, we consider a caplet’s and floorlet’s maturity as being one time step before the actual payment date with a payoff equal to the present value of the actual payment one time step later. The caplet’s payoffs are illustrated in Figure 23.4. We next show how to price this caplet, first with an example and then in general .

FIGURE 23.4: A Caplet Evolution Time 0

Time 1

Example c(1)u = max 0,

0.0395 – 0.04 =0 1.0395

c(1)d = max 0,

0.0465 – 0.04 = 0.0062 1.0465

c(1)u = max 0,

R(1)u – k 1 + R(1)u

c(1)d = max 0,

R(1)d – k 1 + R(1)d

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c(0) = 0.003

Symbols

c(0)

EXAMPLE 23.6: Pricing a Caplet ■

As in the binomial model of Chapter 17, we construct a portfolio consisting of a zero-coupon bond and mma to replicate the traded caplet’s payoffs in both the up and down states. The cost of constructing this synthetic caplet is the arbitrage-free price of the traded caplet.

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■

Using the zero-coupon bond price evolution in Figure 23.2, we can compute the values for the spot rate at time 1 in the up and down states, R(1)u and R(1)d , respectively: R (1)u = 1/B (1, 2)u − 1 = 1/ (0.9620) − 1 = 0.0395 R (1)d = 1/B (1, 2)d − 1 = 1/ (0.9555) − 1 = 0.0465

The Caplet’s Time 1 Payoffs ■

Consider the caplet of Figure 23.4 maturing at time 1 with strike rate k = 0.04 per year. In the up state, the caplet ends out-of-the-money with zero value, as the following computation shows: cu = max [(R (1)u − k) / (1 + R (1)u ) , 0] = max [(0.0395 − 0.04) / (1.0395) , 0] = 0

(23.16a)

In the down state, the caplet ends up in-the-money with value cd = max [(R (1)d − k) / (1 + R (1)d ) , 0] = max [(0.0465 − 0.04) / (1.0465) , 0] = $0.0062

(23.16b)

Constructing the Synthetic Caplet ■

Form a portfolio using m shares of the two-period zero-coupon bond with price B(0, 2) and n units of the mma with value A(0). The cost of construction is

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V (0) ≡ mB (0, 2) + nA (0) ■

(23.17a)

At time 1, we want to find (m, n) to match the portfolio’s value to the traded caplet’s value in each state. This yields two equations in two unknowns: V (1)u ≡ mB (1, 2)u + nA (1) = c (1)u or m0.9620 + n1.0309 = 0

In the up state ∶

In the down state ∶

V (1)d ≡ mB (1, 2)d + nA (1) = c (1)d or m0.9555 + n1.0309 = 0.0062

Subtracting Equation 23.17c from Equation 23.17b gives m × (0.9620 − 0.9555) = −0.0062 or m = −0.9556 Plugging m’s value into Expression 23.17b and solving we get n=

−m0.9620 = 0.8917 1.0309

(23.17b)

(23.17c)

THE SINGLE-PERIOD MODEL

■

So by going short 0.9556 of the two-period zero-coupon bond and buying 0.8917 units of the mma, we can construct a synthetic caplet. The cost of construction is V (0) ≡ mB (0, 2) + n = −0.9556 × 0.93 + 0.8917 = $0.003 This is the arbitrage-free price of the traded caplet.

■

One can repeat this construction using the mma and the three-period zero. Try it and see that since the term structure evolution is arbitrage-free, both approaches give the same caplet price.

The Hedge Ratio (the Holy Grail) The holdings in the risky bond, m, is called the caplet’s delta or the hedge ratio, the holy grail of option pricing. It is the ratio of change in the caplet’s price to a change in the zero-coupon bond’s price. Looking at Equation 23.17b and 23.17c, we see that if one holds the traded caplet and shorts m units of the two-period zero-coupon bond, the resulting portfolio will be riskless and earn the mma’s rate of return.

An Arbitrage Opportunity

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Suppose that the market price of the caplet is $0.005. Then, an arbitrage opportunity exists. The arbitrage opportunity is to sell the traded call for $0.005 and buy the synthetic call at a cost of $.003. At time 1, the payoffs to the short traded call and the long synthetic call are equal in magnitude and opposite in sign, so the net payoff is zero.

We can formalize this example to obtain some general conclusions regarding our method for pricing caplets.

THE CAPLET’S TIME 1 PAYOFFS First, we note that the caplet’s time 1 payoffs are (see Figure 23.4) In the up state ∶

In the down state ∶

R(1)u − k c (1)u = max 0, [ 1 + R(1)u ] R(1)d − k c (1)d = max 0, [ 1 + R(1)d ]

(23.18a)

(23.18b)

CONSTRUCTING THE SYNTHETIC CAPLET We can create a synthetic caplet with cost of construction V (0) = mB (0, 2) + nA (0) with m shares of the two-period zero worth B(0,2) and n units of the mma priced at A(0) = 1.

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Next, at time 1, we match this portfolio’s value with that of the caplet’s: In the up state ∶

V(1)u = mu (0, 2) B (0, 2) + n [1 + R (0)] = c (1)u

In the down state ∶

V(1)d = md (0, 2) B (0, 2) + n [1 + R (0)] = c (1)d (23.19b)

(23.19a)

Solving for m and n yields m=

c (1)u − c (1)d [u (0, 2) − d (0, 2)] B (0, 2)

(23.20a)

u (0, 2) c (1)d − d (0, 2) c (1)u (23.20b) [u (0, 2) − d (0, 2)] [1 + R (0)] To avoid arbitrage, the cost of constructing the synthetic caplet portfolio must equal the traded caplet’s price. Substitution of m and n into this equation and simplification, therefore, gives the arbitrage-free price of the caplet: n=

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c (0) = mB (0, 2) + n u (0, 2) c (1)d − d (0, 2) c (1)u c (1)u − c (1)d B (0, 2) + = [u (0, 2) − d (0, 2)] B (0, 2) [u (0, 2) − d (0, 2)] [1 + R (0)] (23.21) Notice that this caplet’s value does not depend on the actual probability q(0); it only depends on the up and down factors for the two-period zero [u(0, 2), d(0, 2)], the two-period zero’s price B(0, 2), and the spot rate of interest R(0). This is similar to what we found in the binomial option pricing model in Chapter 17. This is an important observation. It implies that if two investors agree on these inputs, but disagree on the actual probability of going up, q(0), they will still agree on the caplet’s price. Of course, disagreement about the actual probability of going up q(0) might lead to a disagreement about the initial zero’s price B(0, 2) given fixed [u(0, 2), d(0, 2), R(0)]. However, if agreement about the zero’s price B(0, 2) occurs, then the caplet’s value is fixed by our no-arbitrage argument, and the two investors will agree on the caplet’s price. This omission occurs because in the derivation of the caplet’s price, we use exact replication. No matter to which node of the tree the zero’s price moves, the synthetic caplet’s payoffs exactly duplicate the traded caplet’s payoffs. Hence we are indifferent to what the actual probability is of moving up or down. Finally, we note that the synthetic construction illustrated uses the two-period bond and the mma. An analogous synthetic construction could have been done using the three-period bond and the mma. This construction generates different holdings in the three-period bond and mma, as distinct from those in the two-period bond and mma, but the arbitrage-free price is identical. This is guaranteed by the equal pseudo-probability condition given by Equation 23.12.

THE SINGLE-PERIOD MODEL

The Hedge Ratio (the Holy Grail) The method for constructing the synthetic caplet also yields the method for hedging the risk of a long caplet position. Determining this hedge ratio, of course, is the holy grail of option pricing (see Chapter 17). If one shorts m shares of the two-period zero for each caplet held long, the resulting portfolio is riskless. The holdings in the risky bond, m, is called the caplet’s delta or hedge ratio. To see why the portfolio is riskless, we just need to rearrange the terms in Equation 23.21 to obtain the time 0 hedged caplet portfolio: c (0) − mB (0, 2) Its payoffs at time 1 are In the up state ∶

c (1)u − mu (0, 2) B (0, 2) = c (1)u − =

In the down state ∶

u (0, 2) c (1)d − d (0, 2) c (1)u = n [1 + R (0)] [u (0, 2) − d (0, 2)]

c (1)d − md (0, 2) B (0, 2) = c (1)d − =

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c (1)u − c (1)d u (0, 2) [u (0, 2) − d (0, 2)]

c (1)u − c (1)d d (0, 2) [u (0, 2) − d (0, 2)]

u (0, 2) c (1)d − d (0, 2) c (1)u = n [1 + R (0)] [u (0, 2) − d (0, 2)]

In both the up and down states, the portfolio has the same value and is therefore riskless. As shown, it earns the spot rate of interest. The hedge ratio, m, given by Equation 23.20a therefore removes all of the interest rate risk from the hedged caplet portfolio.

Risk-Neutral Valuation We can rearrange Equation 23.21 for the caplet’s arbitrage-free price to obtain another insight. Some algebra yields c (0) =

1 + R (0) − d (0, 2) u (0, 2) − [1 + R (0)] 1 c (1)u + c (1)d ( u (0, 2) − d (0, 2) ) ] [1 + R (0)] [( u (0, 2) − d (0, 3) )

Using the definition of the pseudo-probability in Result 23.1 under the no-arbitrage condition, we can rewrite this equation abstractly as c (0) =

𝜋 (0) c (1)u + [1 − 𝜋 (0)] c (1)d 1 + R (0)

(23.22a)

The caplet’s arbitrage-free price is seen to be its expected discounted value using the pseudo-probabilities. This gives a present value formula. The pseudo-probabilities adjust for the risk of the caplet’s payoff, and therefore one can use the riskless rate

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to discount to the present. As in Chapter 17, if one uses this equation to value the caplet, then it is called risk-neutral valuation. Using familiar abstract notation, we can summarize this result (see Result 23.3).

RESULT 23.3 Risk-Neutral Valuation for a Caplet A caplet with strike rate k and maturing at time 1 is given by the formula c (0) =

E𝜋 [c (1)] 1 + R (0)

(23.22b)

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where E𝜋 [ . ] denotes expectation using the pseudo-probability 𝜋(0), R(0) is the time 0 spot rate, and B(1, T) is the T maturity zero-coupon bond’s price at time 1.

As noted earlier, the caplet’s value does not depend on the actual probability of moving to the up state. It only depends on the pseudo-probability. Because of Equation 23.22b, the pseudo-probability is often called the risk-neutral probability. We note in passing that this more abstract equation will prove useful in deriving a put–call parity theorem for caplets and floorlets analogous to that for ordinary calls and puts. Risk-neutral valuation greatly simplifies the computation of the caplet’s arbitragefree price, as the following example shows (see Example 23.7).

EXAMPLE 23.7: Risk-Neutral Valuation Using the binomial tree in Figure 23.2 and the caplet payoffs from Figure 23.4, we get c (0) =

𝜋 (0) c (1)u + [1 − 𝜋 (0)] c (1)d 0.5 × 0 + 0.5 × 0.0062 = = 0.003 1.0309 1 + R (0)

As illustrated, using the risk-neutral valuation method is much easier than constructing the synthetic caplet and solving the two equations for the two unknowns—and it gives the same value, which should not be a surprise!

Valuing a Floorlet Risk-neutral valuation can also be used to value a floorlet. Alternatively, the price of a floorlet can be obtained from that of a caplet by using a put–call parity relationship. This is given by the following result (see Result 23.4).

THE SINGLE-PERIOD MODEL

RESULT 23.4 Caplet–Floorlet Parity [f (0, 1) − k] B (0, 2) = c (0) − p (0)

(23.23)

where time 0 is today, time 1 is the maturity date, f (0,1) is the time 0 forward rate for date 1, B(0,2) is today’s price of a zero-coupon maturing at time 2, and c(0) and p(0) are today’s prices of a caplet and a floorlet, respectively, with identical terms and conditions including strike rate k and maturity date 1. This relation is proved in the appendix to this chapter. Caplet–floorlet parity shows that a position long a caplet and short a floorlet equals a long position in [f (0,1) – K] zero-coupon bonds, represented by the left side of this equation. Given the value of a caplet, the value of the floorlet follows directly from this equation. The next Example 23.8 illustrates the use of caplet–floorlet parity for valuing a floorlet.

EXAMPLE 23.8: Valuing a Floorlet ■

Consider a floorlet that has strike rate K = 0.04 per year and maturity date 2. Its time 2 payoff is p (2) = max [k − R (1) , 0]

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■

The floorlet is a European put option on the spot rate of interest. Note that the floorlet’s payoff at time 2 is based on the spot rate at time 1. Because this payoff is known at time 1, the present value of this payoff at time 1 is obtained by discounting by the spot rate: p (1) =

max [k − R (1) , 0] 1 + R (1)

As with the caplet, since the floorlet’s payoff is known at time 1, using our convention, we consider time 1 as the maturity of the floorlet. ■

Let the term structure evolution be as in Figure 23.2 and Example 23.3. Consider a caplet with the same maturity and the same strike rate as the floorlet’s. We determined the time 0 price of this caplet as c(0) = 0.003 dollars in Example 23.7. The forward rate at time 0 for date 2 is given by Equation 23.4 as f (0, 1) =

■

B (0, 1) 0.97 −1= − 1 = 0.0430 0.93 B (0, 2)

Hence, using caplet–floorlet parity, we have that p (0) = c (0) − [f (0, 1) − k] B (0, 2) = 0.003 − (0.0430 − 0.04) 0.93 = $0.0002 which is the floorlet’s value at time 0. You can check to see that using risk-neutral valuation gives the same answer, as it should!

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Valuing Various Interest Rate Derivatives The method we have developed for pricing caplets via risk-neutral valuation is surprisingly general. It can be applied to any interest rate derivative to obtain its arbitrage-free price. The only modification needed is to replace the caplet’s time 1 payoff with that of the alternative interest rate derivative. We will show how to apply this technique to price American options, swaptions, and futures contracts after studying the two-period model in Chapter 24. We cannot study them here because in a single period model, the complexity of these securities disappear: futures contracts become forward contracts, American options become European options, and swaptions become deterministic.

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Multiple Factors The single-period binomial HJM model is called a one-factor model because at each node of the tree, only one of two possibilities can happen: up and down. A two-factor model has three branches at each node in the tree: up, middle, and down; a three-factor model has four branches at each node, and so forth. A factor corresponds to an economic shock affecting the evolution of the term structure of interest rates. Such shocks are due to the natural randomness in an economy resulting from the ordinary economic activities of production, consumption, technological advances, population growth, and government regulatory policy (monetary and fiscal). One can understand these random shocks by thinking about how to simulate an evolution through an HJM tree. For a one-factor tree, starting at time 0 and progressing forward, flipping a single coin at each node generates a path through the tree. Of course, the coin would need to be biased with the probability of heads being the actual probability of moving up. A coin is the simplest random generating device one can think of. Because only one random generating device is needed to create a path through a binomial tree, it is called a one-factor model. For a two-factor model, at each node in the tree, one needs to flip two coins to decide which branch to take. Recall that a two-factor model has three branches at each node. One coin is needed to decide between up and “the remaining two branches.” A second coin is needed to choose between the remaining two branches: middle or down. Hence you need two of these simplest random generating devices to simulate an evolution through a two-factor HJM tree. Each random generating device is a “factor.” The same logic extends for three or more factors. The valuation and hedging methods for multifactor models are straightforward extensions of those illustrated for the one-factor model. The only difference is that more branches need to be considered, and hedging requires more zero-coupon bonds at each time step because there is more risk (see Jarrow 2002b).

SUMMARY

When modeling the evolution of the term structure of interest rates, one should include enough factors to capture all of the relevant randomness affecting the economy. Macroeconomics can help in understanding the sources of this randomness, as discussed earlier. Researchers have estimated multifactor models and discovered that at least three or four factors are needed to provide term structure evolutions consistent with historical experience.

23.6

Summary

1. The basic equity derivatives are forwards, futures, calls, and put options. The basic interest rate derivatives are forwards (called forward rate agreements), futures, call options (called caplets), and put options (called floorlets). Caplets and floorlets do not trade, but portfolios of caplets, called caps, and portfolios of floorlets, called floors, trade over-the-counter. 2. The single-period binomial HJM model assumes that the zero-coupon bond prices B(t, T) for T = 1, 2, 3 and t = 0, 1 trade in discrete time and that their prices evolve according to a binomial process. First, if the zero-coupon bond matures the next period, that is, at time 1, then the current bond price goes to $1 regardless of the state. Second, the current T > t + 1 maturity zero-coupon bond price either gets multiplied by an up factor u(t, T) and takes the price u(t, T) × B(t, T) with the actual probability q(t) > 0, or it is multiplied by a down factor d(t, T) and takes the price d(t, T) × B(t, T) with probability 1 – q(t).

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3. The single-period zero-coupon bond price binomial tree is arbitrage-free if and only if there exist pseudo-probabilities 𝜋(0) strictly between 0 and 1 such that 𝜋 (0) = 𝜋 (0, T)

for T = 2, 3

where 𝜋 (0, T) =

1 + R (0) − d (0, T) [ u (0, T) − d (0, T) ]

4. Consider a caplet maturing at time 2 with a strike rate of K. The present value of its payoff at time 1 is c(1) = max [0, (R(1) – k)/(1 + R(1))]. Because this payoff is known at time 1, we consider the caplet’s maturity as time 1, and not time 2. The arbitrage-free price of a caplet can be determined using risk-neutral valuation c(0) = E𝜋 [c(1)]/[1 + R(0)], where E𝜋 [·] denotes expectation using the pseudoprobabilities. The hedge ratio m = [c(1)u – c(1)d ]/{[u(0,2) – d(0,2)]B(0,2)} can be used to form a riskless portfolio consisting of a long position in the caplet and shorting this number of two-period zero-coupon bonds. 5. Caplet–floorlet parity is [f (0, 1) − k] B (0, 2) = c (0) − p (0) where f (0, 1) is the time 0 forward rate for date 1, B(0, 2) is the time 0 zerocoupon price with maturity at time 2, and both the caplet with time 0 price c(0) and the floorlet with time 0 price p(0) have strike price k and maturity 1.

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23.7

Appendix

The Equal Pseudo-probability Condition As with the binomial model for pricing equity options (see Chapter 17), we can trade two assets to synthetically construct the price of a third asset. Closely related to this synthetic construction is the equivalence of the pseudo-probabilities, which the following proof shows.

Proof of Necessary Condition ■

We first show how to construct the evolution of the traded two-period zerocoupon bond’s price using a portfolio of the mma and the three-period zerocoupon bond.

■

The payoffs to the traded two-period zero at time 1 in the up and down states are u(0, 2)B(0, 2) and d(0, 2)B(0, 2), respectively.

■

Let us construct a portfolio consisting of n units of the mma and m shares of the three-period bond with time 0 value V (0) ≡ mB(0, 3) + nA(0) such that in both the up and down states, the portfolio’s value matches the price of the two-period zero, that is, V(1)u = mB(1, 3)u + nA (1) = u (0, 2) B (0, 2) and V(1)d = mB(1, 3)d + nA (1) = d (0, 2) B (0, 2)

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■

Using the notation from Figure 23.2, we can rewrite this system of equations as mu (0, 3) B (0, 3) + n [1 + R (0)] = u (0, 2) B (0, 2) mu (0, 3) B (0, 3) + n [1 + R (0)] = d (0, 2) B (0, 2) Solving this set of equations gives m=

[u (0, 3) d (0, 2) − u (0, 2) d (0, 3)] B (0, 2) [u (0, 2) − d (0, 2)] B (0, 2) and n = [u (0, 3) − d (0, 3)] B (0, 3) [u (0, 3) − d (0, 3)] [1 + R (0)]

Substituting these back into V (0) gives the cost of construction as V (0) =

[u (0, 2) − d (0, 2)] B (0, 2) [u (0, 3) d (0, 2) − u (0, 2) d (0, 3)] B (0, 2) B (0, 3) + A (0) [u (0, 3) − d (0, 3)] B (0, 3) [u (0, 3) − d (0, 3)] [1 + R (0)]

Algebra yields 1 + R (0) − d (0, 3) u (0, 3) − (1 + R (0)) B (0, 2) u (0, 2) + B (0, 2) d (0, 2) ( u (0, 3) − d (0, 3) ) ( u (0, 3) − d (0, 3) ) V (0) = 1 + R (0)

APPENDIX

Finally, using Equation 23.9c, we can write the cost of construction as V (0) = ■

𝜋 (0, 3) B (0, 2) u (0, 2) + [1 − 𝜋 (0, 3)] B (0, 2) d (0, 2) 1 + R (0)

(1)

No arbitrage implies that the cost of constructing the synthetic two-period zero must equal the price of the traded two-period zero, that is, B (0, 2) =

𝜋 (0, 3) B (0, 2) u (0, 2) + [1 − 𝜋 (0, 3)] B (0, 2) d (0, 2) 1 + R (0)

(2)

Equation 2 is the third necessary condition (see Equation 23.10).

Proof of Sufficiency ■

To we note that Equation 23.12 implies that Equation 23.13b holds. We use this condition subsequently.

■

Let condition 23.12 hold: 𝜋(0) = 𝜋(0, T) for T = 2, 3 in our interest rate binomial tree. We suppose that an arbitrage opportunity exists and then generate a contradiction of Equation 23.12, thereby proving that the evolution is arbitrage-free.

■

Consider an arbitrage portfolio consisting of a units of the mma, b units of the two-period zero, and c units of the three-period zero-coupon bond. Then, by the definition of an arbitrage trading strategy this portfolio has a zero initial investment:

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a + bB (0, 2) + cB (0, 3) = 0 ■

(3)

After one period, the arbitrage portfolio’s values must be non-negative always and one must be strictly positive, that is: In the up state ∶ In the down state ∶

aA (1) + bB(1, 2)u + cB(1, 3)u ≥ 0 aA (1) + bB(1, 2)d + cB(1, 3)d ≥ 0

where at least one of the last two inequalities is strictly positive. ■

Dividing the last two equations by A(1) and taking expectations using the pseudoprobability 𝜋(0) gives B (1, 2) B (1, 3) E𝜋 a + b +c >0 ( A (1) A (1) )

(4)

But using Equation 23.13b on this last equation (which follows from Equation 23.12) yields a contradiction of Equation 3: a + bB (0, 2) + cB (0, 3) > 0

(5)

Thus, when unique pseudo-probabilities exist, we cannot have an arbitrage opportunity in the tree.

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Proof of Caplet and Floorlet Parity To prove caplet and floorlet parity in the single-period model, consider a caplet and floorlet both with maturity 1 and strike rate k. Their time 2 payoffs are c (2) = max [0, R (1) − k] and p (2) = max [0, k − R (1)] where R(1) is the spot rate. It is easy to show that the following equation holds: R (1) − k = max [0, R (1) − k] − max [0, k − R (1)] = c (2) − p (2)

(6)

One can easily verify that this identity is true by testing values for the spot rate that are greater than or less than the common strike rate k. Add and subtract 1 on the left side and divide by the mma’s time 2 value, A(2). This gives [1 + R (1)] /A (2) − (1 + k) /A (2) = max [0, R (1) − k] /A (2) − max [0, k − R (1)] /A (2) or [1 + R (1)] /A (2) − (1 + k) /A (2) = c (2) /A (2) − p (2) /A (2) (7) Take expectations of (7) with respect to the pseudo-probability to get

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E𝜋

p (2) 1 + R (1) c (2) 1 − (1 + k) E𝜋 = E𝜋 − E𝜋 [ A (2) ] [ A (2) ] [ A (2) ] [ A (2) ]

(8)

The risk-neutral valuation method implies that B (0, 2) = E𝜋

E𝜋

1 [ A (2) ]

1 + R (1) 1 = E𝜋 = B (0, 1) [ A (2) ] [ A (1) ]

(9a)

(9b)

c (0) = E𝜋

c (2) [ A (2) ]

(9c)

p (0) = E𝜋

p (2) [ A (2) ]

(9d)

Substitution of Equations 9a–9d in Equation 8 yields the result B (0, 1) − (1 + k) B (0, 2) = c (0) − p (0) but the left side of this equation equals B (0, 1) − 1 B (0, 2) − kB (0, 2) = [f (0, 1) − k] B (0, 2) [ B (0, 2) ]

(10)

QUESTIONS AND PROBLEMS

where f (0,1) is obtained from the definition of forward rate (see Equation 23.4). Substitution yields the final result: [f (0, 1) − k] B (0, 2) = c (0) − p (0)

23.8

(11)

Cases

Spencer Hall (A) (European Case Clearing House Case 9A95B045). The case ana-

lyzes issues surrounding the hedging of interest rate risk with instruments like interest rate swaps, caps, and collars. Spencer Hall (B) (European Case Clearing House Case 9A97N004). An extension

of the previous case (Spencer Hall [A]), the case considers issues surrounding the cancellation of an interest rate cap by accepting a premium payment. Markborough Properties Inc. (Ivey Publishing Case 9A87B002, European Case

Clearing House Case). The case examines how a company can hedge its interest rate risk by using derivatives like interest rate swaps and caps.

23.9

Questions and Problems

23.1. Show that the following tree is not arbitrage-free.

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FIGURE 1: Zero-coupon Bond Price Evolution Time 0

Time 1

Zero-Coupon Bond Prices

q(0) = 0.3 B(0, 3) = 0.83 B(0, 2) = 0.93 B(0, 1) = 0.97

B(1, 3)u = 0.9167 B(1, 2)u = 0.9620 B(1, 1)u = 1

(“Up” state)

B(1, 3)d = 0.8977 B(1, 2)d = 0.9555 B(1, 1)d = 1

(“Down” state)

R(0) = 0.0309

Money Market Account (mma) A(0) = 1

A(1) = [1 + R(0)] = 1.0309

23.2. Compute the forward rate evolution implied by Figure 23.2 (this is identical

to the preceding tree with B[0,3] = 0.88).

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23.3. In the caplet example 23.6, construct the synthetic caplet using the three-

period zero and mma (the caplet has a strike rate of k = 0.04 or 4 percent). Show that the cost of construction is 0.003, the same as that with the twoperiod zero-coupon bond. 23.4. In the floorlet example 23.7, value the floorlet using risk-neutral valuation

and show you get the same answer as in the text. 23.5. In the caplet example 23.6, suppose the traded caplet has a market price of

$0.002. What is a trading strategy that will generate an arbitrage opportunity? 23.6. In the caplet example 23.6, suppose you have a long position in the caplet.

Form a hedged portfolio of the caplet and the two-period zero-coupon bond that has no interest rate risk. 23.7. Use Figure 23.2. Consider a European call option on the three-period zero-

coupon bond with maturity time 1 and strike price of k = $0.90. Compute the value using risk-neutral valuation. 23.8. Use Figure 23.2. Consider a European put option on the 3-period zero-

coupon bond with maturity time 1 and strike price of K = $0.90. Compute the value using risk-neutral valuation.

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FIGURE 2: Zero-coupon Bond Price Evolution B(1, 3)u = 0.932788 B(1, 2)u = 0.973744 B(1, 1)u = 1 B(0, 3) = 0.897710 B(0, 2) = 0.942596 B(0, 1) = 0.970874 B(1, 3)d = 0.916495 B(1, 2)d = 0.968004 B(1, 1)d = 1

23.9. For the single period evolution in this figure, compute the spot rates at time 0

and at time 1 in both the up and down states. Prove that this tree is arbitragefree (ignore round errors beyond two decimal places in the computation of the pseudo-probabilities). 23.10. For the single-period evolution in the preceding figure, consider a caplet with

maturity time 1 with strike rate k = 0.03. a. What is the value of this caplet at time 0? b. What is the delta for this caplet in terms of the three-period zero-coupon

bond?

QUESTIONS AND PROBLEMS

23.11. For the single-period evolution in the preceding figure, what is the value of a

floorlet with maturity time 1 and strike rate 0.03? Compute the value using risk-neutral valuation. 23.12. For the caplets and floorlets in examples 23.10 and 23.11, show that they

satisfy caplet-floorlet parity. 23.13. For the single-period evolution given in the preceding figure, consider a

European call option with maturity date 1 and strike price K = $0.92 on the three-period zero-coupon bond. What is the arbitrage-free price of this call option? 23.14. For the single-period evolution given in the preceding figure, consider an

American call option with maturity 1 and strike price K = $0.92 on the three-period zero-coupon bond. What is the arbitrage-free price of this call option? Is early exercise (at time 0) optimal? 23.15. For the single-period evolution given in the preceding figure, consider a

European put option with maturity 1 and strike price K = $0.92 on the three-period zero-coupon bond. What is the arbitrage-free price of this put option? 23.16. For the single-period evolution given in the preceding figure, consider a

European put option with maturity date 1 and strike price K = $0.94 on the three-period zero-coupon bond. What is the arbitrage-free price of this put option?

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23.17. For the single-period evolution given in the preceding figure, consider an

American put option with maturity date 1 and strike price K = $0.94 on the three-period zero-coupon bond. What is the arbitrage-free price of this put option? Is early exercise (at time 0) optimal? 23.18. Why do you think caps and floors trade as portfolios of caplets and floorlets in

interest rate markets, whereas in equity markets, calls and puts directly trade? 23.19. Why is it that Eurodollar futures and forward rate agreement rates are not

equal? 23.20. When one compares forward rate agreement contracts to Eurodollar futures

contracts, which have greater counterparty risk? Do you think this might influence the difference between FRA and futures rates? If yes, why?

591

24 Multiperiod Binomial HJM Model 24.1 Introduction

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24.2 The Assumptions 24.3 The Two-Period Model Arbitrage-Free Evolutions Zero-Coupon Bond Prices and Martingales Caplet Pricing Floorlet Pricing Valuing Various Interest Rate Derivatives Multiple Factors

24.4 The Multiperiod Model

24.5 Forwards, Futures, and Swaptions Forward Rate Agreements Eurodollar Futures Comparing Forward and Futures Rates Swaptions

24.6 Summary 24.7 Appendix Linking the Binomial HJM Model and the HJM Libor Model

24.8 Cases 24.9 Questions and Problems

THE ASSUMPTIONS

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24.1

Introduction

The single-period Heath–Jarrow–Morton (HJM) model is useful as a teaching tool to introduce the economic logic underlying the pricing and hedging methodology. However, even as a teaching tool, the single-period model has its limitations. Many interest rate derivatives need at least two periods to capture their complexity. For example, in a single-period setting, an American call option trivializes to a European call, a futures contract simplifies to a forward, and a swaption sheds its uncertainty and becomes a deterministic security! Therefore, even for pedagogical reasons, one needs to extend the HJM model to a multiperiod setting. More important, the real limitation of a single-period model is that just as in the binomial option pricing model of Chapter 18, the term structure evolution only becomes realistic when the time periods become small and the number of periods becomes large. To obtain a model useful for practice, one must extend the HJM model to a multiperiod setting. The two-period model of this chapter starts us along this path. The multiperiod extension just involves more complicated computations as the number of periods increase, easily done on a computer, once the economic logic of the two-period model is mastered (see Jarrow 2002b). This chapter presents the multiperiod binomial HJM model. We start with the term structure of interest rate evolution in a two-period setting. Next, the twoperiod HJM valuation model is presented. The extension to multiple periods and the approximation to the continuous time HJM libor model, the topic of Chapter 25, is discussed. Finally, the multiperiod model is applied to the pricing of caplets, floorlets, forward rate agreements (FRAs), Eurodollar futures, and swaption contracts. For ease of presentation, the material parallels the presentation used in the multiperiod binomial model of Chapter 18. The next chapter presents a continuous time analogue to the Black–Scholes–Merton model for pricing caplets and floorlets.

24.2

The Assumptions

We retain assumptions A1–A5, which have been with us since Chapter 17, repeated here for convenience: A1. A2. A3. A4. A5.

No market frictions No credit risk Competitive and well-functioning markets No intermediate cash flows No arbitrage opportunities

We extend assumption A6 to two periods. A6. Two-period binomial process. The zero-coupon bond prices B(t, T) for t = 0, 1, 2 and T = 1, 2, 3 trade in discrete time, and their prices evolve according to a binomial process. At each time in the tree, there are two states “up” and “down.” First, if the zero-coupon bond matures in the next period, that is, T = t + 1, then the current bond price goes to $1 regardless of the state. Second, standing at time t, the current T > t + 1 maturity zero-coupon bond price either gets multiplied by

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the factor u(t, T) in the up state and takes the price u(t, T) × B(t, T) with the actual probability q(t) > 0 or it is multiplied by the factor d(t, T) in the down state and takes the price d(t, T) × B(t, T) with probability 1− q(t). We assume that u(t, T) > d(t, T). Note that the verbal description is almost identical to that in the single-period model. The only difference is in the number of time steps considered. The twoperiod evolution is given in Figure 24.1. It is the same as the single-period evolution up to time 1 in both the up and down states. The difference occurs after time 1. For this reason, when describing the evolution, we focus only on discussing what happens after the first period.

EXAMPLE 24.1: The Two-Period Term Structure Evolution

Zero-Coupon Bond Prices ■

If we are in the up state at time 1, then zero-coupon bond prices can go up again, or they can go down. The actual probability of going up is q(1)u = 0.2. The prices of the zeros in the up and down states are identified by the subscripts uu and ud, respectively. - Consider the longest-maturity zero-coupon bond. If the up state occurs, then this zero’s price is B(2, 3)uu = 0.9555. If the down state occurs, then the price of this zero is B(2, 3)ud = 0.9504. - Consider the two-period bond. If either state occurs, the bond pays off a dollar. Thus B(2, 2)uu = B(2, 2)ud = 1.

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■

If we are in the down state at time 1, the actual probability of going up is q(1)d = 0.6. The prices of the zeros in the up and down states are identified by the subscripts du and dd, respectively. - Consider the three-period zero. If the up state occurs, then this zero’s price is B(2, 3)du = 0.9440. If the down state occurs, then the zero’s price is B(2, 3)dd = 0.9349. - Consider the two-period bond. If either state occurs, the bond pays off a dollar. Thus B(2, 2)du = B(2, 2)dd = 1.

■

Note that over the time period (1, 2), there is one less zero-coupon bond trading in the term structure of zeros. This is because the one-period bond at time 0 matures at time 1 and disappears from the market.

■

We note that the binomial tree, unlike that in the binomial model of chapter 17, does not recombine to form a lattice. This is because this example is based on a discrete approximation to the HJM libor model that does not recombine (see the appendix to this chapter).

The lower part of Figure 23.3 gives the formalization of this example using the symbols introduced in assumption A6.

Zero-Coupon Bond Prices ■

If we are in the up state at time 1, and if up occurs again, then the three-period bond’s price is u(1, 3)u B(1,3)u = B(2, 3)uu . If down occurs instead, then the threeperiod bond’s price is d(1, 3)u B(1, 3)u = B(2, 3)ud . The two-period bond pays $1 regardless of the state at time 2.

THE ASSUMPTIONS

■

If we are in the down state at time 1, and if up occurs again, then the three- period bond’s price is u(1, 3)d B(1, 3)d = B(2, 3)du . If down occurs instead, then the threeperiod bond’s price is d(1, 3)d B(1, 3)d = B(2, 3)dd . The two-period bond pays $1 regardless of the state at time 2.

The binomial tree in its most general form does not recombine to form a lattice. It could be designed to do so. Subcases of the HJM model where the tree recombines to form a lattice include the Ho and Lee (1986) and Vasicek (1977) models.

The Spot Rate ■

If we are in the up state at time 1, then the spot rate is R(1)u = [1 − B(1, 2)u ] /B(1, 2)u = (1 − 0.9620) /0.9620 = 0.0395 or 3.95 percent

■

(24.1a)

If we are in the down state at time 1, then the spot rate is R(1)d = [1 − B(1, 2)d ] /B(1, 2)d = (1 − 0.9555) /0.9555 = 0.0465 or 4.65 percent

(24.1b)

The Money Market Account ■

The money market account (mma) earns the spot rate each period:

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A (2) = [1 + R (0)] ×

=

1 + R (1)u if up { 1 + R (1)d if down

1.0717 if up { 1.0788 if down

(24.2)

At this point, it is easy to understand how to extend the two-period model to three or more periods. Starting at time 0, the number of zero-coupon bonds trading at each node should exceed the number of periods in the model by at least one. Then, one just adds more time steps. At each time step, the shortest- maturity bond matures, pays $1, and then disappears from the tree. Otherwise, the tree’s structure is no different than the two-period evolution.

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FIGURE 24.1: Two-Period HJM Evolution Time 0

Time 1

Time 2

Zero-Coupon Bond Prices (Example) q(1)u = 0.2

q(0) = 0.3 B(0,3) = 0.88 B(0,2) = 0.93 B(0,1) = 0.97

B(1,3)u = 0.9167 B(1,2)u = 0.9620 B(1,1)u = 1

B(2,3)uu = 0.9555 B(2,2)uu = 1

R(1)u = 0.0395 B(2,3)ud = 0.9504 B(2,2)ud = 1

R(0) = 0.0309 q(1)d = 0.6 B(1,3)d = 0.8977 B(1,2)d = 0.9555 B(1,1)d = 1

B(2,3)du = 0.9440 B(2,2)du = 1

R(1)d = 0.0465 B(2,3)dd = 0.9349 B(2,2)dd = 1

Zero-Coupon Bond Prices (Symbols) Time 0

Time 1

Time 2

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q(1)u u(0,3)B(0,3) = B(1,3)u u(0,2)B(0,2) = B(1,2)u B(1,1)u = 1

q(0) B(0,3) B(0,2) B(0,1)

u(1,3)uB(1,3)u = B(2,3)uu B(2,2)uu = 1

R(1)u d(1,3)uB(1,3)u = B(2,3)ud B(2,2)ud = 1

R(0) q(1)d d(0,3)B(0,3) = B(1,3)d d(0,2)B(0,2) = B(1,2)d B(1,1)d = 1

u(1,3)dB(1,3)d = B(2,3)du B(2,2)du = 1

R(1)d d(1,3)dB(1,3)d = B(2,3)dd B(2,2)dd = 1

Time 0

Time 1

Time 2

Money Market Account (mma) Earns R(0)

A(0) = 1

A(1) = [1 + R(0)] = 1.0309

Earns R(1) [1 + R(0)] × [1 + R(1)u] = A(2)u = 1.0717 [1 + R(0)] × [1 + R(1)d] = A(2)d = 1.0788

THE TWO-PERIOD MODEL

24.3

The Two-Period Model

The two-period binomial HJM model is analogous to the single-period model, once one employs the backward induction argument used in the multiperiod binomial option pricing model of chapter 18. The term structure evolution for the two-period economy is given in Figure 24.1 and was previously described. Let us introduce some simplifying notation. At time 2, we see there are four states possible corresponding to the branches (up, up), (up, down), (down, up), and (down, down). We write the prices at time 2 as B(2, T)su and B(2, T)sd , where s ∈ {u, d} (which is a compact way of writing that s takes on either the value u or d) and T = 2, 3. For example, if s = d and T = 3, then the first expression takes the value B(2,3)du . This is the time 2 price of a three-period zero-coupon bond at the node reached by going down at time 1 and then up at time 2. Figure 24.1 shows that this zero has a price of 0.9440. Notice that the time paths are read from left to right: in du, the letter d denotes that the first movement is down, and the next letter u denotes that the second movement is up.

Arbitrage-Free Evolutions

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Just as in the single-period model, we need to determine necessary and sufficient conditions for the two-period binomial tree to be arbitrage-free.

NECESSARY AND SUFFICIENT CONDITIONS It is clear that if the tree is arbitrage-free, then at each node and time, the single-period evolution at that node must be arbitrage-free. Thus Result 23.1 must hold at every node in the tree— and it turns out that if Result 23.1 holds at every node in the tree, then the tree is arbitrage-free. The proof of this last statement is identical to the sufficiency proof for Result 23.1 in chapter 23 and is therefore omitted. Consequently, we have obtained the following result.

RESULT 24.1 Equal Pseudo-Probability Condition The two-period zero-coupon bond price evolution is arbitrage-free if and only if there exists real numbers 𝜋(0), 𝜋(1)u , 𝜋(1)d strictly between 0 and 1 such that the pseudo-probabilities at any node must be identical, that is, 𝜋 (0) = 𝜋 (0, T) for T = 2, 3

(24.3a)

𝜋(1)s = 𝜋(1, 3)s for s ∈ {u, d}

(24.3b)

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where 𝜋(0, T) = [1 + R(0) – d(0, T)]/[u(0, T) – d(0, T)]; 𝜋(1, 3)s = [1 + R(1)s − d(1, 3)s ]/[u(1, 3)s – d(1, 3)s ]; R(0) and R(1)s are the time 0 and time 1 spot rates, respectively; and u(0, T), d(0, T), u(1, 3)s , d(1, 3)s are the dollar returns on the zero-coupon bond in the up and down states at times 0 and 1, respectively.

In this characterization result, the no-arbitrage condition at time 0 is the same as in the single-period model. Note that at time 2, the last period in the tree, there is only one pseudo-probability. This is because there is only one risky zero-coupon bond trading over the last time step. Hence, over the last time step, the no-arbitrage condition is the same as in Chapter 17. The pseudo-probability just needs to be strictly between zero and one. When considering models with more than two time steps, both of these observations will always be true. First, for every time step except the last, there will be multiple pseudo-probabilities, all of which must be equal and strictly between zero and one for these nodes to be arbitrage-free. Second, over the last time step, there is only one pseudo-probability that must be strictly between zero and one. The next example shows that this arbitrage-free condition is satisfied by the twoperiod binomial evolution in Figure 24.1.

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EXAMPLE 24.2: An Arbitrage-Free Two-Period Binomial Tree This example verifies that the term structure evolution in Figure 24.1 is arbitrage-free. ■

First, we have already checked the node at time 0 in Example 23.4 and verified that it is arbitrage-free. Hence we only need to check the two nodes at time 1 in the up and down states. We need to compute the up and down returns on the three-period zero-coupon bond for all possible states at time 1. These are as follows: u (1, 3)u = B (2, 3)uu /B (1, 3)u = 0.9555/0.9167 = 1.0423 d (1, 3)u = B (2, 3)ud /B (1, 3)u = 0.9504/0.9167 = 1.0367 u (1, 3) d = B (2, 3)du /B (1, 3)d = 0.9440/0.8977 = 1.0516 d (1, 3)d = B (2, 3)dd /B (1, 3)d = 0.9349/0.8977 = 1.0415

■

Then, using expression (24.3), we compute the pseudo-probabilities: 1 + R(1)u − d(1, 3)u 1.0396 − 1.0367 = = 0.5 ( u(1, 3)u − d(1, 3)u ) ( 1.0423 − 1.0367 ) 1 + R(1)d − d(1, 3)d 1.0465 − 1.0415 𝜋(1, 3)d = = = 0.5 ( u(1, 3)d − d(1, 3)d ) ( 1.0516 − 1.0415 )

𝜋(1, 3)u =

THE TWO-PERIOD MODEL

Since this is the last time step and each of these pseudo-probabilities is between zero and one, the tree is arbitrage-free. We denote the value of these pseudo-probabilities as 𝜋(1)u and 𝜋(1)d in the up and down states, respectively.

Zero-Coupon Bond Prices and Martingales Assuming that each node in the tree is arbitrage-free, just as in the single-period model, we can show that the zero-coupon bond prices must equal their expected discounted $1 payoff, using the pseudo-probabilities. To do this, we use backward induction. Standing at time 1 in state s ∈ {u, d}, because the node is arbitrage-free, by definition of 𝜋(1)s , algebra yields 𝜋(1)s B(2, T)su + [1 − 𝜋(1)s ] B(2, T)sd

B(1, T)s =

1 + R(1)s

for T = 2, 3

(24.4a)

or, more abstractly,

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B(1, T)s =

E𝜋1 [B (2, T)] 1 + R(1)s

for T = 2, 3

(24.4b)

This is analogous to the single-period model, but viewed from time 1 in state s. Next, stepping back to time 0, the logic used for the one-period model shows that B (0, T) =

𝜋 (0) B(1, T)u + [1 − 𝜋 (0)] B(1, T)d for T = 2, 3 1 + R (0)

(24.5a)

Substitution of expression (24.4a) into this expression and algebra yields B (0, T) = 𝜋 (0) 𝜋(1)u

B(2, T)uu [1 + R (0)] [1 + R(1)u ]

+𝜋 (0) [1 − 𝜋(1)u ] + [1 − 𝜋 (0)] 𝜋(1)u

B(2, T)ud [1 + R (0)] [1 + R(1)u ] B(2, T)du [1 + R (0)] [1 + R(1)d ]

+ [1 − 𝜋 (0)] [1 − 𝜋(1)u ] or, more abstractly, we get our final result.

B(2, T)dd [1 + R (0)] [1 + R(1)d ]

(24.5b)

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RESULT 24.2 Risk-Neutral Valuation for Zero-Coupon Bond Prices The time 0 value of the zero-coupon bonds can be written as B (0, 2) = E𝜋

1 ( [1 + R (0)] [1 + R (1)] )

(24.6a)

B (0, 3) = E𝜋

B (2, 3) ( [1 + R (0)] [1 + R (1)] )

(24.6b)

and

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where E𝜋 (⋅) denotes expectation using the pseudo-probabilities [𝜋(0), 𝜋(1, s)]; R(0) and R(1) are the time 0 and 1 spot rates, respectively; and B(2,3) is the three-period zero-coupon bond’s price at time 2.

These two expressions show that the zero-coupon bond’s price is its expected discounted $1 payoff using the pseudo-probabilities. Result 24.2 proves that the local expectations hypothesis, introduced in chapter 21 (see Extension 21.2), holds under the pseudo-probabilities and not the actual probabilities. The local expectations hypothesis states that all zero-coupon bonds earn the same expected return over any time period. Essentially, all zerocoupon bonds are equally risky. We see that the local expectations hypothesis is true under the pseudo-probabilities, and as seen later, because the pseudo-probabilities differ from the actual probabilities via a risk adjustment, the local expectations hypothesis cannot hold under the actual probabilities. Last we note that the denominators in expressions (24.6a) and (24.6b) correspond to the time 2 value of the mma A(2). As such, this immediately implies that the price of a zero-coupon bond normalized by the price of the mma, B(t, T)/A(t), is a martingale under the pseudo-probabilities, generalizing the same result proven in the single-period model. The next example applies risk-neutral valuation to zero coupon bonds.

EXAMPLE 24.3: Risk-Neutral Valuation for Zero-Coupon Bonds Using the two-period evolution given in Figure 24.1, we can check to see if the risk-neutral valuation formula holds for the zero-coupon bonds maturing at time 2. The formula is 𝜋 (0) 𝜋 (1)u

B(2, 2)uu [1 + R (0)] [1 + R(1)u ] + [1 − 𝜋 (0)] 𝜋 (1)u

+ 𝜋 (0) [1 − 𝜋 (1)u ] B(2, 2)du

[1 + R (0)] [1 + R(1)d ]

B(2, 2)ud [1 + R (0)] [1 + R(1)u ]

THE TWO-PERIOD MODEL

+ [1 − 𝜋 (0)][1 − 𝜋 (1)u ]

B(2, 2)dd

[1 + R (0)] [1 + R(1)d ] 1 1 + 0.25 = 0.25 [1.0309] [1.0396] [1.0309] [1.0396] 1 1 +0.25 + 0.25 [1.0309] [1.0465] [1.0309] [1.0465] = 0.97 = B (0, 2) validating the result in the two-period example. As shown, the price of the three-period zero is equal to its value in the four states at time 2 multiplied by the pseudo-probability of each state occurring and discounted to time 0 using the spot rates of interest. This is expression (24.6b) written out in longhand.

Caplet Pricing We are now ready to price a caplet in this two-period economy. Although they are not shown on the two-period tree, the spot rate at time 2 in each state can be computed as 1 − B(2, 3)us R(2)us = (24.7a) B(2, 3)us and

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R(2)ds =

1 − B(2, 3)ds for s ∈ {u, d} B(2, 3)ds

(24.7b)

Consider a caplet maturing at time 3 with strike rate k. The caplet’s time 3 payoff is c(3) = max[0, R(2) −k]. This value is known at time 2, hence the time 2 present value of this payoff is R (2) − k (24.8) c (2) = max 0, [ 1 + R (2) ] Using the convention introduced in the previous chapter, for valuation purposes, we will consider the caplet’s maturity as time 2. These are the payoffs used in the tree to compute the time 0 value of this caplet.

BACKWARD INDUCTION Let us recall the (fictional) super sleuth Sherlock Holmes, who popularized the method of backward induction: start at the end, work backward through time, eventually solving the problem at the beginning. To price the caplet using this method of backward induction, we pretend we are standing at time 1 and value the caplet at that time. Then, having the time 1 values, we move back to time 0 and value the caplet at that date, which gives our answer. METHOD 1: CONSTRUCTION OF THE SYNTHETIC CAPLET AT TIME 1 Standing at time 1 at state s 𝜖 {u, d}, we want to construct the portfolio of the three-period zero and mma account such that the payoffs to the portfolio match the payoffs to the traded caplet. The method is identical to that used in the

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one-period model, except the notation is now different. We choose n shares of the three-period zero and m shares of the mma with cost V(1)s = mB(1, 3)s + n A (1) such that at time 2, in the up state: V(2)su = mu(1, 3)s B(1, 3)s + nA(1)[1 + R(1)s ] = c(2)su (24.9a) in the down state: V(2)sd = md(1, 3)s B(1, 3)s + nA(1)[1 + R(1)s ] = c(2)sd (24.9b) Solving for m and n yields m=

n=

c(2)su − c(2)sd [u(1, 3)s − d(1, 3)s ] B(1, 3)s

(24.10a)

u(1, 3)s c(2)sd − d(1, 3)s c(2)su

(24.10b) [u(1, 3)s − d(1, 3)s ] A (1) [1 + R(1)S ] Substitution of m and n into the cost of construction and simplification gives the arbitrage-free price of the caplet: c (1)s = mB (1, 3)s + nA (1)

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=

c(2)su − c(2)sd [u(1, 3)s − d(1, 3)s ] B (0, 2)

B (0, 2) +

u(1, 3)s c(2)sd − d(1, 3)s c(2)su

(24.11a)

[u(1, 3)s − d(1, 3)s ] [1 + R(1)s ]

METHOD 2: RISK-NEUTRAL VALUATION AT TIME 1 Using the arbitragefree condition and the pseudo-probabilities, algebra yields c (1)s =

𝜋(1)s c (2)su + [1 − 𝜋(1)s ] c (2)sd 1 + R(1)s

or, more abstractly, c (1)s =

E𝜋1 [c (2)] 1 + R(1)s

(24.11b)

(24.11c)

We note again that this is the same method we used in the single-period model, except the time and state notation has changed. This gives a present value formula. The caplet’s time 1 value is its discounted expected payoff at time 2 using the pseudoprobabilities.

PRICING THE CAPLET AT TIME 0 Having priced the caplet at time 1 in both the up and down states, we now move back to time 0 to price the caplet. Note that this is the same problem solved in the single-period model, except that we replace the previous payoffs with expression (24.11c). The arbitrage-free price of the caplet is therefore 𝜋 (0) c(1)u + [1 − 𝜋 (0)] c(1)d (24.12a) c (0) = 1 + R (0)

THE TWO-PERIOD MODEL

Substitution of the caplet’s time 1 values from expression (24.11b) and algebra yields c (0) = 𝜋 (0) 𝜋(1)u

c (2)uu [1 + R (0)] [1 + R(1)u ]

+ 𝜋 (0) [1 − 𝜋(1)u ] + [1 − 𝜋 (0)] 𝜋(1)u

c (2)ud [1 + R (0)] [1 + R(1)u ] c (2)du

(24.12b)

[1 + R (0)] [1 + R(1)d ]

+ [1 − 𝜋 (0)] [1 − 𝜋(1)u ]

c (2)dd [1 + R (0)] [1 + R(1)d ]

Using more abstract notation, we can alternatively write this as our next result.

RESULT 24.3 Risk-Neutral Valuation Formula for a Caplet A caplet with strike rate k and maturity date 2 is given by

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c (0) = E𝜋

c (2) ( [1 + R (0)] [1 + R (1)] )

(24.13)

where E𝜋 (⋅) denotes expectation using the pseudo-probabilities [𝜋(0), 𝜋(1, s)]; R(0) and R(1) are the time 0 and 1 spot rates, respectively; and c(2) is the caplet’s value at time 2.

Expression (24.13) illustrates risk-neutral valuation for pricing a caplet, and it gives a present value formula! Today’s price is the expected discounted value of the caplet’s time 2 payoff, where the expectation uses the pseudo-probabilities. The pseudoprobabilities adjust for risk, consequently discounting is with the riskless spot rate of interest. For the purposes of computing the arbitrage-free price of a caplet, expression (24.13) is easy to use, as illustrated in the next example.

EXAMPLE 24.4: Pricing a Caplet This example uses the risk-neutral valuation method to price a caplet with maturity time 2, strike rate k = 0.05, and notional of $1.

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■

First, we need to compute the caplet’s payoffs at time 2 in all states. To do this, we must first compute the spot rate of interest at time 2 in all states. Using Figure 24.1, we get 1 − B(2, 3)uu B(2, 3)uu 1 − B(2, 3)ud R(2)ud = B(2, 3)ud 1 − B(2, 3)du R(2)du = B(2, 3)du 1 − B(2, 3)dd R(2)dd = B(2, 3)dd

R(2)uu =

■

1 − 0.9555 0.9555 1 − 0.9504 = 0.9504 1 − 0.9440 = 0.9440 1 − 0.9349 = 0.9349 =

= 0.0465 = 0.0522 = 0.0593 = 0.0696

The caplet’s payoffs at time 2 are

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R(2)uu − k 0.0465 − 0.05 = max [0, c (2)uu = max 0, ]=0 [ 1 + R(2)uu ] 1.0465 R(2)ud − k 0.0522 − 0.05 c (2)ud = max 0, = max [0, ] = 0.0021 [ 1 + R(2)ud ] 1.0522 R(2)du − k 0.0593 − 0.05 c (2)du = max 0, = max [0, ] = 0.0088 [ 1 + R(2)du ] 1.0593 R(2)dd − k 0.0696 − 0.05 c (2)dd = max 0, = max [0, ] = 0.0183 [ 1 + R(2)dd ] 1.0696 These payoffs are illustrated in Figure 24.2. ■

Finally, using the risk-neutral valuation formula, the caplet’s time 0 value is c (0) = 𝜋 (0) 𝜋(1)u

c (2)uu

[1 + R (0)] [1 + R(1)u ] c (2)ud +𝜋 (0) [1 − 𝜋(1)u ] [1 + R (0)] [1 + R(1)u ] c (2)du + [1 − 𝜋 (0)] 𝜋(1)u [1 + R (0)] [1 + R(1)d ] c (2)dd + [1 − 𝜋 (0)] [1 − 𝜋(1)u ] [1 + R (0)] [1 + R(1)d ] 0.0021 0 + 0.25 = $0.25 [1.0309] [1.0396] [1.0309] [1.0396] 0.0088 0.0183 +0.25 + 0.25 [1.0309] [1.0465] [1.0309] [1.0465] = $0.0068

THE TWO-PERIOD MODEL

Floorlet Pricing The same method can be applied to price a floorlet. The arbitrage-free valuation formula will be expression (24.13), with the payoff of the floorlet replacing that of the caplet. Alternatively, given the price of the caplet, one can use caplet–floorlet parity to value the caplet. The next result gives caplet–floorlet parity in the twoperiod setting.

RESULT 24.4 Caplet–Floorlet Parity [f (0, t − 1) − k] B (0, t) = c (0) − p (0)

(24.14)

where f (0, t − 1) is the time 0 forward rate for date t − 1, B(0, t) is the time 0 zero-coupon bond price with maturity at time t, and both the caplet with time 0 price c(0) and the floorlet with time 0 price p(0) have strike rate k and maturity date t − 1 for t = 2, 3.

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The proof is identical to that of the single-period model and is therefore omitted (see the appendix to Chapter 23). The next example shows how to price a floorlet using caplet–floorlet parity.

FIGURE 24.2: Caplet Evolution Time 0

Time 1

Time 2 c(2)uu = max 0,

0.0465 – 0.05 =0 1.0465

c(2)ud = max 0,

0.0522 – 0.05 = 0.0021 1.0522

c(2)du = max 0,

0.0593 – 0.05 = 0.0088 1.0593

c(2)dd = max 0,

0.0696 – 0.05 = 0.0183 1.0696

c(1)u

c(0) = 0.0068

c(1)d

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EXAMPLE 24.5: Valuing a Floorlet ■

Consider a floorlet that has strike rate k = 0.05 and maturity time 2. Its time 3 payoff is p (3) = max [k − R (2) , 0]

■

The floorlet is a put option on the spot rate of interest. Note that the floorlet’s payoff at time 3 is based on the spot rate at time 2. Because this payoff is known at time 2, the present value of this payoff at time 2 is obtained by discounting by the spot rate. For valuation purposes, this is the floorlet’s maturity date. max [k − R (2) , 0] p (2) = 1 + R (2)

■

Let the term structure evolution be as in Figure 24.1. Consider a caplet with the same maturity and the same strike price as the floorlet’s. We determined the time 0 price of this caplet as c(0) = $0.0068 in Example 24.5.

■

The forward rate at time 0 for date 2 is f (0, 2) =

B (0, 2) 0.93 −1= − 1 = 0.0568 0.88 B (0, 3)

Hence, using caplet–floorlet parity (see expression [24.14]), we have that

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p (0) = c (0) − [f (0, 2) − k] B (0, 3) = 0.0068 − (0.0568 − 0.05) 0.88 = 0.0008 The floorlet has a value of $0.0008 at time 0 on a notional of $1.

Valuing Various Interest Rate Derivatives The same approach used for pricing a caplet can be used to price any interest rate derivative. The only modification is to replace the payoffs of the caplet with those of the interest rate derivatives under consideration. For American-type interest rate derivatives, just as in Chapter 18, in the backward induction argument, one needs to make a decision regarding early exercise or not at each node of the tree. Otherwise, the procedure is identical to that illustrated earlier.

Multiple Factors The extension to multiple factors in the two-period model is the same as in the single-period model. For example, a two-factor model has three branches at each node in the tree. The pricing methodology extends easily, see Jarrow (2002b) in this regard.

FORWARDS, FUTURES, AND SWAPTIONS

24.4

The Multiperiod Model

The extension of the binomial HJM model to more than two time periods is computationally demanding but straightforward in terms of the ideas and concepts. When building the tree, more zero-coupon bonds are needed in the term structure as the time horizon in the model grows. There should be at least one more zerocoupon bond in the term structure than there are periods in the model. Except for the extended size of the bond price vectors at each node, the analysis is identical to that already presented. As the time progresses in the tree, one zero-coupon bond matures and leaves the term structure. When valuing interest rate derivatives, just as in the two-period model, backward induction is used. All of the previous results hold in a T-period model (see Jarrow [2002b] for more on this topic). The multiperiod binomial HJM model can be used to approximate a continuous time HJM model by making the time steps small in size. The link between the multiperiod binomial HJM model and the HJM libor model discussed in the next chapter is provided in the appendix to this chapter. The appendix shows how to choose the up and down factors in the zero-coupon bond price evolution so that the implied simple forward rates have a lognormal distribution in the limit.

24.5

Forwards, Futures, and Swaptions

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Let us revisit FRAs, Eurodollar futures, and swaptions in the context of the twoperiod binomial model, although the results hold more generally in a multiperiod setting. As noted in the introduction to this chapter, the complexities of these contracts are only evidenced in multiperiod models.

Forward Rate Agreements We previously discussed FRAs in Chapter 21. Recall that in our discrete time model, the FRA payoff at time T can be written as LN × [R (T − 1) − fFRA (0, T − 1)]

(24.15)

where LN is the notional value of the contract and fFRA (0, T− 1) is the time 0 FRA rate for time T− 1. The payment on the FRA occurs one period after the uncertainty in the contract’s payoff is determined. Consider an FRA with maturity date 3 on a notional value of $1. Using the previous expression, we have that the FRA’s time 3 payoff is R (2) − fFRA (0, 2)

(24.16)

A long position in the FRA receives at time 3 the difference between the spot rate and the agreed-on FRA rate fFRA (0,2). At time 0, the market-clearing FRA rate is determined such that the value of the FRA is zero. It was shown in Chapter 21 that

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this market-clearing condition implies that the FRA rate is equal to the forward rate, that is, B (0, 2) fFRA (0, 2) = f (0, 2) = −1 (24.17) B (0, 3) The forward rates at time 0 are easily computed from the zero-coupon bond prices at time zero in the two-period evolution of Figure 24.1: ⎡f (0, 2) = 0.0568⎤ ⎢f (0, 1) = 0.0430⎥ ⎢ ⎥ ⎣f (0, 0) = 0.0309⎦

Eurodollar Futures We previously discussed Eurodollar futures contracts in Chapter 21 as well. Recall that in our discrete time model, a futures contract with delivery date T on a notional value of LN has a futures price equal to

and

F (t) = LN × [1 − ifut (t)] at time t

(24.18a)

F (T) = LN × [1 − R (T)] at time T

(24.18b)

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where ifut (t) is the time t futures rate. The cash flows over the life of the futures contract are determined by markingto-market. We have that the cash flows at times t = 1, 2, 3,.. are [F (t) − F (t − 1)] = −LN × [ifut (t) − ifut (t − 1)]

(24.18c)

Consider a Eurodollar futures contract with maturity 2 on a notional value of $1. We can easily determine the arbitrage-free futures price in our two-period HJM model.

ARBITRAGE-FREE FUTURES PRICES We use backward induction in conjunction with the risk-neutral valuation formula to determine the arbitrage-free futures prices. Standing at time 1, using the risk-neutral valuation formula, the value of the futures contract is E𝜋1 [F (2) − F (1)] V (1) = 1 + R (1) where the 1 denotes taking the expectation at time 1. The time 1 futures price F( 1) is set such that the contract has zero value. This implies that E𝜋1 [F (2) − F (1)] 0= 1 + R (1) Some simple algebra yields the result F (1) = E𝜋1 [F (2)]

(24.19)

FORWARDS, FUTURES, AND SWAPTIONS

The time 1 futures price is seen to be equal to the expected time 2 futures price, where the expectation is computed using the pseudo-probabilities. Last, moving back to time 0, again using the risk-neutral valuation formula, the value of the futures contract is V (0) =

E𝜋 [F (1) − F (0)] 1 + R (0)

As before, the time 0 futures price F(0) is set such that the contract has zero value. This implies that E𝜋 [F (1) − F (0)] 0= 1 + R (0) or, equivalently, (24.20) F (0) = E𝜋 [F (1)] Similarly, the time 0 futures price is equal to the expected time 1 futures price under the pseudo-probabilities. Expressions (24.19) and (24.20) prove the fact that futures prices are a martingale under the pseudo-probabilities.

ARBITRAGE-FREE FUTURES RATES We can now determine the arbitragefree futures interest rates using expressions (24.18a) and (24.18b). Substitution of these expressions into the arbitrage-free futures prices gives

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1 − ifut (1) = E𝜋1 [1 − R (2)] and 1 − ifut (0) = E𝜋 [1 − ifut (1)] or ifut (1) = E𝜋1 [R (2)] and ifut (0) = E𝜋 [ifut (1)]

(24.21)

The futures rates are also seen to be martingales under the pseudo-probabilities. One last computation gives us our final result. Using the notation of Figure 24.1, we can write expression (24.21) as ifut (1) = 𝜋(1)s R(2)su + [1 − 𝜋(1)s ] R(2)sd

(24.22a)

for s ∈ {u, d}. Then, substitution of this expression into (24.21) gives ifut (0) = 𝜋 (0) {𝜋 (1)u R (2)uu + [1 − 𝜋 (1)u ] R (2)ud } + [1 − 𝜋 (0)] {𝜋 (1)d R (2)du + [1 − 𝜋 (1)d ] R (2)dd }

(24.22b)

Writing this expression more abstractly yields the following result.

RESULT 24.5 Eurodollar Futures Rates Consider a Eurodollar futures contract with maturity date 2 on a notional value of $1. The market clearing futures rate is ifut (0) = E𝜋 [R (2)]

(24.23)

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CHAPTER 24: MULTIPERIOD BINOMIAL HJM MODEL

where E𝜋 [⋅] denotes expectation using the pseudo-probabilities [𝜋(0), 𝜋(1,s)] and R(2) is the spot rate at time 2.

As evidenced by this result, the futures interest rate is the expected futures spot rate using the pseudo-probabilities. In contrast to common belief, the futures rate is not the expected future spot rate using the actual probabilities. Because the pseudo- and actual probabilities differ by a risk adjustment, we see that Eurodollar futures rates reflect both risk premia and the actual probabilities, generating a biased estimate of the future spot rate of interest.

Comparing Forward and Futures Rates We can now compare the forward and futures interest rates. Using the definition of the spot rate R(2), we can rewrite expression (24.23) as ifut (0) = E𝜋

1 −1 ( B (2, 3) )

(24.24)

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but 1 ⎛ 1 ⎞ E𝜋 ( ⎜ A (2) ⎟ A (2) ) B (0, 2) 1 ⎟> = E𝜋 ⎜ E𝜋 = = 1 + f (0, 2) ( B (2, 3) ) B (0, 3) ⎜ B (2, 3) ⎟ B (2, 3) 𝜋 ⎝ A (2) ⎠ E ( (2) ) A The first strict inequality is due to a well-known result in probability called Jensen’s inequality (given a random variable x, E [1/x] > 1/E[x]). The second to last equality uses the risk-neutral valuation formulas for the zero-coupon bonds. The last equality is just the definition of the forward rate. Substitution of this insight into expression (24.24) shows that ifut (0) > f (0, 2) (24.25) The futures rate is strictly greater than the forward rate. Both contracts are betting on the future spot rate of interest R(2). However, interest rate futures contracts have marking-to-market, whereas FRAs do not. This implies that futures contracts have cash flow reinvestment risk that FRAs do not. Hence, to induce a speculator to enter a futures contract at zero value, a rate higher than the FRA rate needs to be included. This reinvestment risk difference between the two contracts is the cause of the inequality in expression (24.25). As stated many times before, futures and forward contracts are fraternal, not identical, twins! The next example illustrates the determination of eurodollar futures rates.

FORWARDS, FUTURES, AND SWAPTIONS

EXAMPLE 24.6: Eurodollar Futures Rates ■

This example computes the time 0 futures rates for a Eurodollar futures contract maturing at time 2 using the term structure evolution in Figure 24.1. Example 24.4 already computed the spot rates at time 2, which are: R(2)uu = 0.0465, R(2)ud = 0.0522, R(2)du = 0.0593, and R(2)dd = 0.0696. Then, ifut (0) = E𝜋 [R (2)] = 0.25 (0.0465 + 0.0522 + 0.0593 + 0.0696) = 0.0569

■

For comparison, the time 0 forward rate for time 2 is fFRA (0, 2) = f (0, 2) =

B (0, 2) 0.93 −1= − 1 = 0.0568 0.88 B (0, 3)

As the theory showed, the futures rate is greater than the corresponding FRA rate.

Swaptions

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A swaption is an option (call or put) on the value of a swap at a future date. As an option, there is a strike price K and maturity date T, and the underlying—the swap— needs to be completely specified, receiving fixed and paying f loating (or conversely). We illustrate the valuation of a swaption using the example in Figure 24.1.

EXAMPLE 24.7: Swaption Valuation To value a swaption, one must first value the underlying swap. ■

Consider a swap receiving floating and paying fixed with a notional of $100 million and maturing at time 2. Let the fixed rate on the swap be 0.04 per year. The floating rate is the spot rate R(t).

■

Using the valuation formula in Chapter 22 (Result 22.1), the swap’s value at time 1 can be written abstractly as the difference between the value of the floating-rate loan less the value of the fixed-rate loan: VSwap (1) = PVFloat (1) − PVFixed (1) = 100 − [100 × (0.04) × B (1, 2) + 100 × B (1, 2)] Here a floating rate loan is always valued at par ($100 million), and the value of a fixed-rate loan is the present value of its coupon and principal payment at time 2. Using the zero-coupon bond prices from Figure 24.1, we obtain VSwap (1)u = 100 [1 − (0.04) 0.9620 − 0.9620] = −0.0475 VSwap (1)d = 100 [1 − (0.04) 0.9555 − 0.9555] = 0.6248

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■

Consider a swaption written on this swap. Formally, let’s consider a swaption that is a European call option on this swap with a maturity date time 1 and strike price K = 0. The payoff on the swaption at its maturity, time 1, is Swaption (1) = max [VSwap (1) , 0] If the swap’s value is greater than zero, the option is exercised and the long gets the underlying swap with value VSwap (1).

■

Using the values of the swap computed earlier, the swaptions’ values at time 1 in the up and down state are Swaption(1)u = max [−0.0475, 0] = 0 Swaption(1)d = max [0.6248, 0] = 0.6248

■

Finally, using the risk-neutral valuation formula, the time 0 value of the swaption is Swaption (0) =

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=

𝜋 (0) Swaption(1)u + [1 − 𝜋 (0)] Swaption(1)d 1 + R (0) 0.5 × 0 + 0.5 × 0.6248 = 0.3030 1.0309

From an economic perspective, a swaption is an insurance policy that enables the long to enter a swap at a later date, at terms determined at time 0. Such a contract would be of interest to a corporate treasurer who entered into a floating-rate loan at time 0, and who may want to switch to a fixed-rate loan at a later date if interest rates increase, but at the fixed rates determined at time 0. If exercised at a later date, his floating-rate loan plus the swap changes the floating-rate loan into a fixed-rate loan, with the time 0 fixed rates—and there is no cost to exercise because the strike price is zero! This is a wonderful insurance product for the corporate treasurer, but as with all insurance contracts, it has a cost—premium—to be paid. The previous example demonstrated how to determine the arbitrage-free value of this premium.

24.6

Summary

1. The binomial HJM model assumes that the zero-coupon bond prices B(t, T) for t = 0, 1, 2 and T = 1, 2, 3 trade in discrete time and that their prices evolve according to a binomial process. First, if the zero-coupon bond matures the next period, that is, T = t + 1, then the current bond price goes to $1 regardless of the state. Second, standing at time t, the current T > t + 1 maturity zerocoupon bond price either gets multiplied by an up factor u(t, T) and takes the price u(t, T) × B(t, T) with the actual probability q(t) > 0 or it is multiplied by a down factor d(t, T) and takes the price d(t, T) × B(t, T) with probability 1− q(t).

APPENDIX

2. The two-period zero-coupon bond price binomial tree is arbitrage-free if and only if there exist pseudo-probabilities 𝜋(0), 𝜋(1)u , 𝜋(1)d strictly between 0 and 1 such that 𝜋 (0) = 𝜋 (0, T) for T = 2, 3 and 𝜋(1)s = 𝜋(1, 3)s for s ∈ {u, d} where 𝜋(0, T) = [1 + R(0) – d(0, T) ] / [u(0, T) – d(0, T) ] and 𝜋(1, 3)s =[1 + R(1)s – d(1, 3)s ] / [u(1, 3)s – d(1, 3)s ] 3. Consider a caplet maturing at time 2 with a strike rate of k. Its payoff at time 2 is c(2) = max {0, [R(2) – K ] / 1 + R(2)]}. The arbitrage-free price of a caplet can be determined using the risk-neutral valuation formula c(0) = E𝜋 (c(2) / ( [1 + R(0)] [1 + R(1)])), where E𝜋 (⋅) denotes expectation using the pseudo-probabilities. 4. Caplet–floorlet parity is [f (0, 1) − k] B (0, 2) = c (0) − p (0) where f (0,1) is the time 0 forward rate for date 1, B(0,2) is the time 0 zero-coupon bond price with maturity at time 2, and both the caplet with time 0 price c(0) and the floorlet with time 0 price p(0) have strike rate k and maturity 1.

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5. Consider a Eurodollar futures contract maturing at time 2. The arbitrage-free futures rate at time 0 is ifut (0) = E𝜋 [R(2)]. The futures rate is always larger than the corresponding FRA rate.

24.7

Appendix

Linking the Binomial HJM Model and the HJM Libor Model The binomial interest rate model is easy to understand and to implement. But is it really useful? The answer is yes. In fact, the binomial model can approximate the popular HJM libor model, which is commonly used by financial institutions to price caps, floors, and swaptions. This appendix shows how to set the parameters of the binomial HJM model so that it approximates the HJM libor model of Chapter 25. With interest rate models, the specification is more complex than that used in the case of the binomial option pricing model in Chapter 18 to approximate the Black–Scholes–Merton model. The logic of the procedures, however, is identical. We provide the convergence parameters for a one-factor HJM model. These parameters are selected such that the simple forward rate i(t, T) converges to a lognormal distribution, which yields the libor model. First, impose a discrete time step of length ∆t. Fix a partition of time such that N∆t = 𝛿. Let T* be the maximum maturity zero-coupon bond under consideration with t ≤ T ≤T*, where T* = t + M𝛿 = t + M ⋅ N ⋅ ∆t. A total of M ⋅ N time steps of length ∆t are in the time horizon.

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The forward rate and spot rate over the time interval [t, t + ∆t] are defined by f (t, T) =

B (t, T) 1 −1 ( B (t, T + Δt − 1) ) Δt

and R (t) =

1 1 −1 ( B (t, t + Δt) ) Δt

We would like to investigate the limit of the multinomial tree as ∆t → 0. We note that the forward rate f (t, T) will converge to the continuously compounded forward rate f (t, T) over the time period [t, t+ dt]. We are interested in the limit distribution of the simple forward rate, defined by i (t, T) =

B (t, T) 1 for 0 ≤ t ≤ T ≤ T∗ −1 )𝛿 ( B (t, T + 𝛿)

where [T, T+ 𝛿] corresponds to the time interval in the future over which the forward rate applies; 𝛿 is a fixed time period, say, 0.25 years. Choose at each time step t (see Jarrow 2002b, p. 286) 𝜋t = 1 > 2

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T−Δt

u (t, T) =

[1 + R (t)] ⎛ T−Δt ⎞ cosh ⎜⎜ 𝛾 (t, j) Δt3/2 ⎟⎟ ∑ ⎝j = t+Δt ⎠

∑

𝛾(t, j)Δt3/2

e j = t+Δt

T−Δt

d (t, T) =

[1 + R (t)] ⎛ T−Δt ⎞ cosh ⎜⎜ 𝛾 (t, j) Δt3/2 ⎟⎟ ∑ ⎝j = t+Δt ⎠

∑

𝛾(t, j)Δt3/2

e j = t+Δt

where cosh(x) ≡ (e−x + ex )/2 and the volatility functions for the continuously compounded forward rates satisfy 𝛿 ⋅ i (t, T) 𝜎 (t, T) ( 1 + 𝛿 ⋅ i (t, T) ) 𝛾 (t, T + 𝛿) = 𝛾 (t, T) + 𝜕T 𝜕

(1)

for the given inputs {B(t, T): T ≥ t} and {𝜎(t, T): T ≥ t}, where 𝜎(t, T) is a deterministic function of time t (see Shreve 2004, p. 442).

CASES

The 𝜎(t, T) correspond to the instantaneous volatilities of the simple forward rates i(t, T) in the libor model. Define X (t, T) = Approximate

𝛿 ⋅ i (t, T) 𝜎 (t, T) ( 1 + 𝛿 ⋅ i (t, T) )

𝜕X (t, T) on the partition by its backward-looking derivative, that is, 𝜕T 𝜕X (t, T) X (t, T − Δt) − X (t, T) ≈ Δt 𝜕T

where X(t, t − ∆t) ≡ 0. The solution to expression (1) is given by the following. Let T = t + j∆t + K𝛿, where j = 0, … , N and K = 1, … , M− 1. Then, for fixed 𝛿, and for each j = 0, … , N− 1, K 𝜕X (t, t + jΔt + k𝛿) (2) 𝛾 (t, T) = 𝛾 (t, t + jΔt) + ∑ 𝜕T k =0

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To specify a process, r( t, t + j∆t) for j = 0, . . . , N − 1 can be arbitrarily specified. We suggest setting them equal to zero. Given are B(t, T), 𝜎(t, T) for T = t + n∆ for n = 0, . . . , MN. From B(t, T), one computes R(t) and i(t, T). From i(t, T), 𝜎(t, T), one computes 𝛾(t, T). Finally, substitution gives u(t, T), d(t, T). Then, the distribution for simple forward rates i(T, T)– i(t, T) converges to a lognormal distribution as ∆t → 0. For implementation, to estimate the parameters, one often chooses 𝜎 (t, T) = 𝜎 (T − t) to be deterministic functions of time to maturity. It is important to note that the binomial tree generated by this approximation is not a lattice. The nonlattice structure results from the desire to obtain lognormally distributed simple forward rates in the limit. To extend this approximation to a multifactor model, see Jarrow (2002b).

24.8

Cases

Liability Management at General Motors (Harvard

Business School Case 293123-PDF-ENG). The case studies issues related to managing the liability structure of General Motors using interest rate derivatives.

Columbia River pulp Company Inc.—Interest rate hedging Strategy (Ivey

Publishing Case 9A90B037, European Case Clearing House Case). The case examines issues related to hedging interest rate risk in a company’s floating rate debt using interest rate derivatives like caps, collars, and swaps. Spencer Hall (A) (European Case Clearing House Case 9A95B045). The case ana-

lyzes issues surrounding the hedging of interest rate risk with instruments like interest rate swaps, caps, and collars.

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24.9

Questions and Problems

24.1. Consider Figure 24.1 where B(0,1) = $0.95 replaces $0.97. Prove that the

new tree is not arbitrage-free. 24.2. In the caplet example 24.4, suppose the strike rate k = 0.06. Value the caplet. 24.3. In the floorlet example 24.5, suppose the strike rate k = 0.06. Value the

floorlet. 24.4. Show that the caplet and floorlet values obtained in problems 24.2 and 24.3

satisfy caplet–floorlet parity. 24.5. Use Figure 24.1. Consider a European call with maturity time 2 and strike

price 0.95 on the three-period zero. What is the value of this call option? 24.6. Use Figure 24.1. Consider a European put with maturity time 2 and strike

price 0.95 on the three-period zero. What is the value of this call option? 24.7. Consider the Eurodollar futures contract in example 24.6. a. Compute the time 1 futures rates in the up and down states s : [ifut (1)

=E𝜋 [R(2)]].

b. Compute the forward rate agreement rate at time 1 for maturity 2 in the up

and down states. Show that these two are the same because both contracts mature in one period. 24.8. For the swap in example 24.7, let the strike price be K = $0.50. Compute

the value of the swap using risk-neutral valuation. Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

24.9. Why are swaptions a useful interest rate derivative? Give an example to explain

your answer. 24.10. Suppose a borrower has a 10-year floating-rate loan but wants a fixed-rate

loan. Explain how the borrower can change the floating-rate loan into a fixedrate loan using caps and floors. 24.11. Suppose the term structures of Eurodollar futures rates are increasing for times

1, 2, 3, and 4. Does this mean that the spot rate of interest is expected to increase over the next four time periods?

QUESTIONS AND PROBLEMS

The next questions are based on this figure.

FIGURE 24. A: Zero-coupon Bond Price Evolution B(2, 3)uu = 0.962057 B(2,2)uu = 1 B(1, 3)u = 0.932788 B(1, 2)u = 0.973744 B(1, 1)u = 1 B(2, 3)ud = 0.953824 B(2, 2)ud = 1

B(0, 3) = 0.897710 B(0, 2) = 0.942596 B(0, 1) = 0.970874

B(2, 3)du = 0.951966 B(2, 2)du = 1 B(1, 3)d = 0.916495 B(1, 2)d = 0.968004 B(1, 1)d = 1 B(2, 3)dd = 0.941612 B(2, 2)dd = 1

24.12. Compute the spot rates of interest at every node in the tree.

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24.13. Is the two-period evolution arbitrage-free? Prove your answer. 24.14. Consider a caplet with maturity date time 2 and strike rate k = 0.03. Compute

the arbitrage-free value of this caplet. 24.15. Consider a floorlet with maturity date time 2 and strike rate k = 0.03. What

is the value of this floorlet? 24.16. Using caplet–floorlet parity, verify the answers in problems 24.14 and 24.15. 24.17. Consider a European call with maturity time 2 and strike price $0.95 on the

three-period zero. What is the value of this call option? 24.18. Consider a European put option with maturity date time 2 and strike price

$0.93 on the three-period zero-coupon bond. What is the value of this put option? 24.19. Consider an American put option with maturity date time 2 and strike price

$0.96 on the three-period zero-coupon bond. What is the value of this put option? 24.20. Compute the Eurodollar futures rates for a contract maturing at time 2 and

compare them to the forward rate agreement rate for a contract maturing at time 3.

617

25

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The Heath–Jarrow–Morton Libor Model 25.1 Introduction

Forward Rates and Martingales

25.2 Why Caplets?

Risk-Neutral Valuation

25.3 A History of Caplet Pricing Black’s Model

Actual versus Pseudo-probabilities

25.8 Caps and Floors

The Heath–Jarrow–Morton Model

Caps

The HJM Libor Model

Floorlets

EXTENSION 25.1 Black’s Model for Pricing Commodity Options and Caplets

Caplet–Floorlet Parity Floors

25.9 Using the HJM Libor Model

25.4 The Assumptions

The Greeks

25.5 Pricing Caplets

Delta Hedging

Caplet and Floorlet Pricing (CFP) Data Caplet Pricing Model

25.6 The Inputs

Gamma Hedging

25.10 American Options, Futures, Swaptions, and Other Derivatives

Observable Inputs

25.11 Summary

The Average Forward Rate Volatility

25.12 Appendix

25.7 Understanding the HJM Libor Model

25.13 Cases 25.14 Questions and Problems

WHY CAPLETS?

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25.1

Introduction

Once upon a time, an expert on interest rate derivatives and his wife went to a bank to open a certificate of deposit that would earn a fixed rate of interest. The bank gave them an option to change the interest rate once during the investment’s lifetime and lock in the then-current rate. After explaining the various terms and conditions, the bank officer asked, “Do you understand this option?” The professor chuckled to himself and was tempted to reply, “Yes, and I even know how to value the option, do you?” But he just said yes, under his wife’s glaring eyes. They finished the formalities and went on their way. As illustrated, even standard banking products contain interest rate derivatives requiring prudent handling! This chapter continues our study of the Heath–Jarrow–Morton (HJM) model by extending the discrete time model of Chapters 23 and 24 to a continuous-time model. You may recall that we assumed no interest rate risk in the continuous-time Black– Scholes–Merton (BSM) model. Now it is back with a vengeance! This chapter shows how to price the most basic interest rate derivative, a caplet, in the continuous-time HJM libor model. Of course, a caplet is just a European call option on an interest rate, the same derivative studied in the BSM model. As such, this chapter is the analogue of Chapters 19 and 20 on the BSM model. The next section explains why pricing caplets is a significant milestone in our study of derivatives. After, we provide a brief history of caplet pricing models. Next are the assumptions underlying the HJM libor model, the caplet pricing formula, and a discussion of the inputs. Continuing, we reintroduce martingales, riskneutral valuation, and the actual and pseudo-probabilities. The presentation is couched in terms of forward rates because of their desirable properties. The pricing of floorlets follows using caplet–floorlet parity. Then, we discuss delta and gamma hedging. The valuation of American options, futures contracts, swaptions, and other interest rate derivatives completes the chapter.

25.2

Why Caplets?

Why does pricing a caplet in continuous time cap (pun intended!) our study of derivatives? ■

Our mantra has been to focus on the key interest rate derivatives: forwards, futures, calls, and puts. Once you understand them well, you can understand most interest rate derivatives by applying the build and break approach (see Chapter 7).

■

We studied forwards in Chapter 21. We discussed futures, calls, and puts on interest rates in Chapters 23 and 24 in a discrete time model. This Chapter repeats the discussion of futures, calls, and puts on interest rates in a continuous time setting.

■

This takes us to caplets, which are call options on interest rates. Once one prices a caplet, a floorlet follows with caplet–floorlet parity. Futures are studied after caplets and floorlets.

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So the time has come for the real stuff! First, let’s take a peek at the history of caplet pricing models.

25.3

A History of Caplet Pricing

Black’s Model The first widely used model for pricing caplets was based on Fischer Black’s (1976) formula for pricing commodity options (see Extension 25.1 for a discussion of the model, and Figure 25.1 for a diagram showing the evolution of these models). Black’s model required the assumption of constant continuously compounded libor spot interest rates. Obviously this is an invalid assumption in a world with random interest rates. Nonetheless, the model was used in the beginning because little else was available.

FIGURE 25.1: Evolution of Caplet Pricing Models

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Black–Scholes– Merton model 1973

Well understood by traders

Black’s model 1976

Black–Scholes–Merton model modified to price forwards and futures options

Black’s model modified to price caplets

Arbitrage within model (spot rate constant and forward rate stochastic)

Heath–Jarrow– Morton model 1992

Arbitrage-free term structure evolution

HJM libor model for pricing caplets 1995, 1997

Assumes lognormally distributed simple forward rates

A HISTORY OF CAPLET PRICING

621

The Heath–Jarrow–Morton Model Early interest rate option pricing models (OPMs), like the early equity OPMs, required the estimation of risk premium. The estimation of risk premium, which is equivalent to estimation of expected returns on zero-coupon bonds, is an impossible task. Just as the BSM model bypassed this thorny issue of risk premium estimation in the context of equity options, the HJM model did the same for interest rate derivatives. Consequently, it fostered the expansion of the swap and interest rate derivatives markets.

The HJM Libor Model Black’s model for pricing caplets admits arbitrage because it is impossible for both continuously compounded spot rates to be constant and simple forward rates to be random. This inconsistency was noted in the original HJM paper (Heath, Jarrow, and Morton 1992). Following this demonstration, a search began to find a stochastic evolution for interest rates where continuously compounded spot rates are random yet forward rates are lognormally distributed. The lognormal distribution was needed to maintain the essence of Black’s formula.

EXTENSION 25.1: Black’s Model for Pricing Commodity Options and Caplets

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In 1976, Black published “The Pricing of Commodity Contracts” in the Journal of Financial Economics. This paper extended the BSM model to price an option on commodity futures. Using notation similar to Result 19.1, Black’s formula for a European commodity futures call option is c = B (0, T) [FN (d1 ) − KN (d2 )]

(1)

where today is time 0, B(0, T) = e−rT is today’s price of a zero-coupon bond that matures at time T, r is the continuously compounded risk-free interest rate per year; T is the common maturity date for both the option and forward (or futures) contracts; F is the forward (or futures) price determined today; N(.) is the cumulative standard normal distribution function, d1 = [log(F /K) + (𝜎2 T / 2)] /𝜎√T and d2 = d1 − 𝜎√T; K is the strike price; and 𝜎 is the volatility of the continuously compounded forward (or futures) price. As noted in part II of the book, forward and futures prices are the same if interest rates are constant. Note that formula (1) can be obtained by replacing the stock price S in the BSM model with FB(0,T). Black’s model easily extends to price options on forward contracts where the option matures at date T but the forward matures at some later date. The new call price formula is the same as in Equation 1, except that the zero-coupon bond price and the forward price correspond to the later date. Next, to apply this model to caplets, one considers the forward rate i(0,T) computed as simple interest (no compounding). Under this forward rate, Black’s formula for pricing a caplet that matures at time T with strike rate k is c = B (0, T + 𝛿) [i (0, T) N (d1 ) − kN (d2 )] 𝛿 (2)

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CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

This is the formula in Equation 1 with i(0,T) replacing F, the volatility of the simple forward rate 𝜎B replacing 𝜎, and an additional term 𝛿 denoting the time interval over which the caplet’s interest payment applies. Note the similarity of this formula to the HJM libor caplet pricing formula of Result 25.1. Unfortunately, Black’s model does not satisfy our assumption A5 of no arbitrage. This happens because the model makes two assumptions that cannot both hold at the same time: first, the spot rate of interest (the riskless rate) is constant, and second, the forward rate is stochastic. These assumptions are mutually inconsistent. If the spot rate is always constant, then the forward rate must be nonrandom as well. Assuming that they hold together generates arbitrage opportunities within the model! This inconsistency was the major motivation for the development of the HJM libor model. By giving an arbitrage-free justification for Black’s formula (with modified inputs), the HJM libor model overcame these difficulties and allowed the pricing of caplets and floorlets.

Klaus Sandmann, Dieter Sondermann, Kristian Miltersen (1995); Kristian Miltersen, Klaus Sandmann, Dieter Sondermann (1997); and Alan Brace, Dariusz Gatarek, Marek Musiela (1997) independently discovered the solution. They came up with the observation that although continuously compounded forward rates cannot be lognormally distributed in an HJM model, simple forward rates can. This special case of the HJM model is known as the libor model. It also goes by the names libor market model, forward libor model, or the market model.

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25.4

The Assumptions

The first five assumptions underlying the HJM libor model are identical to those in Chapter 19, and the sixth assumption is analogous to the lognormal assumption underlying the BSM model. This is no surprise because of the libor model’s lineage to the BSM model via Black’s (1976) model (see Figure 25.1). Recall from the last two chapters that the caplet’s underlying assets are the term structure of Eurodollar deposits (and swaps) or, equivalently, the term structure of Eurodollar zero-coupon bonds. The relevant assumptions from Chapter 19 are assumed to hold for these traded zero-coupon bonds. They are repeated here for convenience (and emphasis). A1. No market frictions. We assume that trading involves no market frictions and that the assets are perfectly divisible. This helps us to create a benchmark model to which one can add frictions, and it’s a reasonable approximation for institutional traders. Furthermore, frictionless markets is even becoming a reasonable first approximation for small traders because the accumulated advances in information technology have drastically reduced trading costs. A2. No credit risk. This is a reasonable assumption for exchange-traded derivatives but may be less so for over-the-counter (OTC) derivatives. In reality, OTC transactions often impose collateral provisions on the counterparties to lower such credit risk. We assume, as a first approximation, that the OTC collateral provisions remove credit risk from these OTC contracts. Interestingly, derivatives subject to

THE ASSUMPTIONS

credit risk are priced by extending this term structure modeling framework, which is studied in Chapter 26. A3. Competitive and well-functioning markets. In a competitive market, traders act as price takers. In a well-functioning market, price bubbles are absent. Despite this being a reasonable assumption, fines have been levied against Barclays bank for manipulating libor index rates during the financial crisis of 2007. A4. No intermediate cash flows. As zero-coupon bonds do not have cash flows before they mature, this assumption is automatically satisfied. A5. No arbitrage opportunities. This assumption is self-explanatory. We discard old assumption A6, no interest rate uncertainty, and we now allow interest rates to fluctuate randomly. In this regard, we need to impose an assumption on the evolution of the term structure of interest rates. To facilitate an understanding of this assumption, let us first revisit the concept of a forward rate from earlier chapters. Recall that Equation 21.9 gives today’s forward rate (for a discrete time interval) for the future period [T, T + 1] as

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f (0, T) =

B (0, T) −1 B (0, T + 1)

(25.1)

where B(0, T) and B(0, T + 1) are today’s (time 0’s) prices for the zero-coupon bonds maturing at times T and (T + 1), respectively. This definition is reasonably general. It holds irrespective of whether interest rates are stochastic or deterministic, but the limitation is that the time interval is discrete and of unit length. We need to generalize this aspect of the definition. The question is, how would you generalize this definition to compute a forward rate over any time period [T, T + 𝛿], even for an infinitesimal time interval? Example 25.1 helps to answer this question.

EXAMPLE 25.1: Forward Rate Computations

Forward Rates (for a Discrete Time Interval) ■

We compute forward rates and report them in Table 25.1. - The first column reports time T: time 0 is today, time 1 is one year, and so on. - The second column reports today’s prices of the zero-coupon bonds, B(0, T), which mature at time T. Their numerical values are reported in the third column. For example, today’s price of a zero that matures after one year is B(0, 1) = $0.97, after two years is B(0, 2) = $0.93, and so on (these numbers are taken from Table 21.3). B(0, 0) = $1 by definition.

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TABLE 25.1: Computing Different Types of Forward Rates (See Example 25.1) Time T Price of a B(0, T) Zero B(0, T) Values

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■

Length of Interval

Forward Rates f (0, 0) ≡ R(0)

Forward Rate Computations B(0, 0) − 1 = 0.0309 B(0, 1)

0

B(0, 0)

1

1

1

B(0, 1)

0.97

1

1.0027

B(0, 1.0027)

0.9699

1/365 = 0.0027

1.5

B(0, 1.5)

0.95

0.5

2

B(0, 2)

0.93

1

f (0, 2)

B(0, 2) − 1 = 0.0568 B(0, 3)

3

B(0, 3)

0.88

1

f (0, 3)

B(0, 3) − 1 = 0.0732 B(0, 4)

f (0, 1) i

1 365

(0, 1)

i0.5 (0, 1)

B(0, 1) − 1 = 0.0430 B(0, 2) B(0, 1) − 1 × 365 = 0.0421 [ B(0, 1.0027) ] B(0, 1) − 1 × 2 = 0.0421 [ B(0, 1.5) ]

We compute forward rates using definition Equation 25.1. - The forward rate for the interval [0, 1] is f (0, 0) = B(0, 0)/B(0, 1) − 1 = 0.0309 and is given in the fourth and fifth cells of the second row. Relation (21.12) tells us that this is also the spot rate, R(0). Several of these computations are also reported in Table 21.3. - The forward rate over the interval [1, 2] is f (0, 1) =

■

B (0, 1) − 1 = 0.0430 B (0, 2)

(25.2a)

We have two dates in the forward rate’s notation: today (time 0), when the rate is quoted, and the date (time 1) when the rate becomes effective. Because we know the ending date (time 2), we skip writing it as a third argument. Although many of our examples consider the length of the time period to be one year, it need not be so in a continuous-time setting.

Simple Forward Rates (in Continuous Time) ■

Suppose you want to compute a forward rate as in Equation 25.2a over a fixed but not a unit time interval such as six months. For this, we use the concept of simple interest, and we consider a simple forward rate. To compute this rate, we need a zero maturing at time 1.5: assume it has a price of B(0, 1.5) = $0.95. Considering the time interval [1, 1.5], we write the simple forward rate as per year. i0.5 (0, 1) =

B (0, 1) 1 0.97 −1 = − 1) 2 = 0.04210526 ) 0.5 ( 0.95 ( B (0, 1.5)

(25.2b)

THE ASSUMPTIONS

Note that in the definition of a simple interest rate, the time interval over which the interest rate applies needs to be indicated. The first time argument in the simple interest rate notation, t = 0, corresponds to the current date, while the second time argument corresponds to the start date of the future time interval T = 1. The subscript 0.5 following the symbol “i” indicates the length of the time interval over which the interest rate applies: half of a year. Finally, note that multiplication by 2 converts the six-month interest rate into an annual rate. ■

Why go back to the simple interest discussed in Chapter 2? First, the Eurodollar market is a market in which the derivatives (such as swaps, caplets, and floorlets) use simple interest rates to compute cash flows, and second, it will enable us to price caplets using a modified Black’s formula.

Continuously Compounded Forward Rates (in Continuous Time) ■

Suppose instead that you want to compute the simple forward rate over an infinitesimal time interval one year into the future, [1, 1 + Δt], as Δt → 0. Let us first approximate this by setting Δt = 1/365 = 0.002740 in Equation 25.2b which is one day measured in years. This requires a zero-coupon bond that matures after one year plus one day with a price of B(0, 1.002740) = $0.969888 today. Then, the simple forward rate over the small time interval Δt = 1/365 is i

1 365

(0, 1) =

B (0, 1) − 1 × 365 = 0.000115 × 365 = 0.04214920 ( B (0, 1.002740) )

(25.2c)

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per year. Notice that multiplication by 365 converts the daily interest rate into an annual rate. ■

Using this rate, if you invest a dollar starting in one year for one day, then you get $1.000115 back in one year and one day.

■

Taking smaller and smaller time intervals enables us to define the continuously compounded forward rate via Equation 25.2c as f ̃ (0, 1) = lim iΔt (0, 1) = lim Δt→ 0

B (0, 1) 1 −1 ) Δt + Δt)

Δt→ 0 ( B (0, 1

(25.2d)

Here, we introduce a different notation for the continuously compounded forward rate, as opposed to a simple interest rate over an infinitesimal time interval. This new notation avoids using a time interval subscript because the length of the time interval to which this forward rate applies is always fixed at an instant.

The example contains two important ideas that we rewrite for emphasis. First, we define the simple forward interest rate. Replacing 1 with T and 0.5 with 𝛿 in Equation 25.2b gives us the simple forward rate over [T, T + 𝛿] (i.e., the forward rate expressed as a simple interest rate). We define this at time 0 for date T as i (0, T) =

B (0, T) 1 −1 × ( B (0, T + 𝛿) ) 𝛿

(25.3a)

Note that we delete the subscript because for the remainder of this chapter, the time interval over which the simple forward rate applies is fixed and always equals to 𝛿. To reduce clutter, we started the clock at time 0, although our analysis holds even if h l k i t

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Second, we define the continuously compounded forward rate by replacing 1 with T and (1+ Δt) with (T + Δt) in Equation 25.2d, where Δt is the time interval over which the interest rate applies. Taking limits as Δt goes to zero gives the continuously compounded forward rate: ˜

f (0, T ) = lim

B (0, T ) 1 −1 ) Δt + Δt)

Δt→0 ( B (0, T

(25.3b)

Now, we add our assumption on the stochastic evolution of simple forward rates. A6. The simple forward rate i (T, T ) over [T, T+ 𝛿] follows a lognormal distri-bution. This assumption is that the natural logarithm of the simple forward rate follows a normal distribution. In symbols, 1 ∫0T 𝜇s ds − 𝜈2T (T) + 𝜈T √T⋅z 2

i (T, T) = i (0, T) e q (z) =

with probability

1 2 − z e 2 /√2𝜋

(25.4a)

where time 0 is today, time T is the maturity date, 𝜇T is the mean percentage change per year in the simple forward rate,

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𝜈2T =

∫0T 𝜎2s ds T

(𝜈 is a Greek letter, pronounced “nu”)

(25.4b)

𝜎s is the time s – maturity simple forward rate [i(0, s)]’s volatility per year, q(z) are the actual probabilities, and z is a standard normal random variable with zero mean and unit variance. Note that the relevant volatility in this evolution is 𝜈T , which is the “average” of the simple s – maturity forward rates’ instantaneous volatilities over the option’s life. The mean is assumed to be a deterministic function of time to maturity T. The s – maturity forward rate’s volatility 𝜎s is assumed to be a constant. This assumption is similar to the BSM model assumption A8 for the stock price’s evolution presented in relationship (19.2). Here, however, the simple forward rate replaces the stock price as the variable with a lognormal distribution. Other assumptions for the evolution of the simple forward rate will generate different term structure evolutions and different caplet pricing models. However, these alternative assumptions are usually imposed on the evolution of the continuously compounded forward rates ˜f (0, T) and not the simple forward interest rates as in the HJM libor model. Some of these alternative models go by the names of the assumed evolution: the Gaussian model, the extended Vasicek model, and the affine model (see Jarrow 2010). These alternative models are left for more advanced courses. We complete our list of assumptions by retaining assumption A7 from Chapter 19 with respect to the trading of the Eurodollar zero-coupon bonds.

PRICING CAPLETS

A7. Trading takes place continuously in time. This assumption is selfexplantory. These assumptions set the stage for introducing the HJM libor model.

25.5

Pricing Caplets

Caplets are European call options on the simple forward rate. For many of our examples in this chapter, we use the following data.

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Caplet and Floorlet Pricing (CFP) Data ■

The caplets and floorlets start trading today (time 0) and expire after one year (time T).

■

The notional principal LN is $100 million.

■

The strike rate (which may be the cap rate or the floor rate) k is 4 percent per year.

■

The payoffs are based on the six-month libor rate index, the simple forward rate i(0,T). Assuming that this six-month period has 182 days, 𝛿 = 182/365 = 0.4986 years is the investment period over which simple interest is computed.

■

The average forward rate volatility 𝜈T is 0.15, or 15 percent per year.

■

If the caplet (or the floorlet) is in-the-money, the payment is made 0.4986 years after the expiration date (at time [T + 𝛿]).

■

The zero-coupon bond prices and simple forward rates are taken from the previous example (see Table 25.1).

Caplet Pricing Model Before introducing the pricing model, let us illustrate the workings of a caplet in continuous time (see Example 25.2).

EXAMPLE 25.2: An Interest Rate Caplet’s Payoffs ■

International Treats and Restaurants Inc.’s (INTRest, a fictitious name) treasurer buys a caplet today (time 0) with maturity T = 1 year, a strike rate of k = 4 percent, and notional of LN = $100 million (see Figure 25.2 for a timeline).

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CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

FIGURE 25.2: Timeline for a Caplet (In Continuous Time)

Start date Time 0 (Today)

Intermediate dates

Maturity date Time T (1 year)

Payment date Time (T + δ) (1.4986 years)

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Time interval, δ = 0.4986 year Buyer and writer negotiate caplet’s terms: the notional LN, the strike rate k, the maturity date T

Long can exercise an American caplet

Long pays premium and becomes owner of the caplet

Traders can close out their positions by reverse trades

libor i(T, T) determined for the period δ.

If i(T, T ) > k, long’s payoff is cT+δ = LN × [i(T, T ) – k] × δ If i(T, T ) ≤ k, caplet expires out-of-the- money

Value of Long’s payoff on this date, cT = B(T, T + δ ) × LN × max[i(T, T ) – k] × δ, where the zero-coupon bond pays $1 at time (T + δ )

Zero-sum game— Long’s gain is Short’s loss

■

Suppose the six-month libor rate index realized after one year is 5 percent. This is the simple forward rate, i(T, T) = i(1, 1) = 0.05.

■

Use the caplet and floorlet pricing (CFP) data: - The caplet’s payoff at time (T + 𝛿) = 1.4986 years is c1,4986 = max (0.05 − 0.04, 0) × (182/365) × 100 million = $498, 630.14

(25.5a)

The payment is received 𝛿 years (a fraction of a year) after the caplet’s maturity at time T. It equals the yearly simple interest rate difference [i(T, T) − k] times the period 𝛿 over which the interest is earned times the notational. - The present value of the payment on the maturity date is obtained by multiplying it by the relevant zero-coupon bond’s price. Suppose B(1, 1.4986) = 0.9850 is the price of a zero-coupon bond at time 1 (at time T) which matures after six more months (at time [T + 𝛿]). Then, the time 1 value of the caplet’s payoff is c1 = 0.9850 × max (0.05 − 0.04, 0) × (182/365) × 100 million = $491, 150.68

(25.5b)

PRICING CAPLETS

The caplet’s payoff from this example can be easily formalized. Consider a caplet with notional principal LN , strike rate k, maturity date T, and payment date (T + 𝛿) and where the time period over which the simple interest rate is computed is 𝛿 years. The caplet’s payoff on the payment date is cT+𝛿 = max [i (T, T) − k] × 𝛿 × LN

(25.6a)

where the simple forward rate i(T, T) is the interest rate at time T that applies over the time period [T, T + 𝛿]. To obtain the value of this payoff at time T, discount this cash flow by multiplying it by the relevant zero-coupon bond’s price. The caplet’s value on the maturity date is cT = B (T, T + 𝛿) max [i (T, T) − k] × 𝛿 × LN (25.6b) where B(T, T + 𝛿) is the time T price of a zero-coupon bond that pays a dollar at time (T + 𝛿). For computation, one can think of time T as both the maturity date and the payoff date of the caplet, but in this case, one needs to use Equation 25.6b as the caplet’s payoff. Under the HJM libor model, it can be shown that the caplet’s time 0 value is given by the following formula (see Result 25.1).

RESULT 25.1 Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

HJM Libor Model Formula for Caplet Pricing A caplet’s price is c = B (0, T + d) [i (0, T) N (d1 ) − kN (d2 )] × 𝛿 × LN

(25.7a)

where B(0, T + 𝛿) is today’s price of a zero-coupon bond that matures at time (T + 𝛿), T is the caplet’s maturity date, 𝛿 is the time period for interest computation, i(0, T) is the simple forward rate per year that applies to the interval [T, T + 𝛿], N( . ) is the cumulative standard normal distribution function (with mean 0 and standard deviation 1),

d1 =

log [ (

𝜈2 × T i (0, T ) + T 2 ] k ) 𝜈T √T

and d2 = d1 − 𝜈T √T

(25.7b)

k is the strike rate (which is the cap rate), 𝜈T is the average forward rate volatility over the caplet’s life, and LN is the notional principal.

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We discuss the proof of this result later in this chapter when discussing risk-neutral valuation. The caplet formula is similar in appearance to the standard BSM formula but applied to the simple forward rate at time T, i(0, T), and not the stock price (which is irrelevant in the world of interest rate derivatives). The key difference from the standard BSM formula is that the stock price is replaced by the bond price B(0, T + 𝛿) multiplied by the simple interest rate i(0, T). This occurs because B(0, T + 𝛿) × i(0, T) corresponds to the present value of i(T, T) dollars received at time (T + 𝛿). In addition to the simple forward rate i(0, T), the caplet’s value depends on the zero-coupon bond price B(0, T + 𝛿), the strike rate k, the maturity date T, and the average forward rate volatility 𝜈T . As with the BSM formula, the mean percentage change in the simple forward rate does not appear in this equation. Had it appeared, the thorny problem of risk premium computation would have remained with us. This “important omission” makes the formula usable. Analogous to the BSM model, this fact implies that if two traders agree on the inputs [i(0, T), B(0, T + 𝛿), k, T, 𝜈T ] but disagree on the mean percentage change in simple forward rates, they will still agree on the caplet’s price. Example 25.3 demonstrates Result 25.1.

EXAMPLE 25.3: Pricing a Caplet

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■

Consider the CFP data. - The caplet is traded today (time 0) and matures at T = 1 year. - The notional principal LN = $100 million and the strike rate k = 4 percent per year. - The contract applies to the six-month libor rate index (which is the simple forward rate). The interest is computed over 182 days, or 𝛿 = 182/365 = 0.4986 year. - The simple forward rate is i(0, T) = i(0, 1) = 0.0421 (see the cell in the fifth row and fifth column in Table 25.1). - The average forward rate volatility 𝜈T is 0.15, or 15 percent per year. - A zero-coupon bond maturing after 1.4986 years is worth B(0,T + 𝛿) = B(0,1.4986) = $0.95 today.

■

Compute d1 and d2 : log d1 =

𝜈2 × T i (0, T) + T ( k ) 2 𝜈T √T

=

log (

0.0421 0.152 × 1 + 0.04 ) 2

= 0.4170 and

0.15√1

d2 = d1 − 𝜈T √T = 0.4170 − 0.15 = 0.2670 ■

Using Excel, we get cumulative standard normal distribution function values: N (d1 ) = N (0.4170) = NORMSDIST (0.4170) = 0.6616 and N (d2 ) = N (0.2670) = NORMSDIST (0.2670) = 0.6052

THE INPUTS

■

The caplet’s price using the HJM libor model formula (25.7a) for a $100 million notional is c = B (0, T + 𝛿) [i (0, T) N (d1 ) − kN (d2 )] × 𝛿 × LN = B (0, 1.4986) [i (0, 1) N (0.4170) − 0.04N (0.2670)] × 0.4986 × 100 million = 0.95 × (0.0421 × 0.6616 − 0.04 × 0.6052) × 0.4986 × 100 million = $172, 842.45

25.6

The Inputs

Collecting inputs for the HJM libor caplet pricing model is analogous to gathering the inputs for the BSM model. Most of the inputs are easily obtained, except for the average forward rate volatility.

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Observable Inputs The easiest inputs to obtain are the strike rate k, expiration date T, and time period for interest computation 𝛿. These are written into the contract. What about the simple forward rate? Had the underlying been a Treasury security, one could observe zerocoupon bond prices from the market—but we are using Eurodollars and libor rate indexes. This difference only presents a minor complication. As the forward rate is equivalent to the FRA rate (see Result 21.3), one can obtain the simple forward rate i(0, T) by observing the rates from FRA contracts trading in the OTC markets. The quoted FRA rates are based on the simple interest rates as defined in Equation 25.3a.1 A Eurodollar zero-coupon bond’s price can be created in several ways from these FRA rates. For example, you can use the FRA rates in the following formula to compute the zero-coupon bond prices: B (0, T) =

1 [1 + i (0, 0) × 𝛿] [1 + i (0, 𝛿) × 𝛿] ⋯ [1 + i (0, T − 𝛿) × 𝛿]

(25.8)

where i(0, 0) is the spot rate that applies over [0, 𝛿], i(0, 𝛿) is the simple forward rate computed over the next time period of length 𝛿, and so on. This is a known relationship (Equation 21.1a of Result 21.2) that has been slightly modified to account for the time intervals not being of unit length, but equal to 𝛿 time units.

1

We point out that when we defined the forward rate in earlier chapters, for simplicity, we always fixed the time interval over which the rates applied to be equal to one unit of time. Here, to correspond to practice, it is necessary to make the time interval differ from one time unit. It is set equal to 𝛿 time units.

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Alternatively, you can obtain these zero-coupon bond prices directly from Eurodollar rates (for maturities up to one year) and swap rates (for maturities beyond one year). The procedure for obtaining these zero-coupon bond prices from swap rates was given in Extension 22.3.

The Average Forward Rate Volatility Like the thorny problem of volatility estimation encountered in the context of the BSM model, the average forward rate volatility is also the hardest input to estimate in the HJM libor model. As before, there are two standard methods for volatility estimation: an explicit method using past simple forward rates, yielding a historical volatility, and an implicit method—calibration—using caplet market prices, yielding an implied volatility.

FIGURE 25.3: Integration and the Average Forward Rate Volatility

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y

y = g(x)

0 a

b

x

s-maturity simple forward rate variance σs2

0

0.25

…

1.25 1.5

Maturity s

THE INPUTS

Note that the equation for the volatility differs from that used for equity options. Here it equals the forward rate’s average volatility over the caplet’s life, that is,

𝜈T = √

∫0T 𝜎2s ds T

(25.9a)

where 𝜎s is the time s – maturity simple forward rate i(0, s)’s volatility per year. To compute this equation, we need to estimate each individual s – maturity forward rate’s volatility 𝜎s . The trick here is that we need to use constant maturity forward rates and compute their volatility just as we did the stock’s return volatility in Chapter 19. Except for these two differences, the methods for estimating the simple forward rate’s volatility per unit time are analogous to those for equity options discussed in Chapters 19 and 20. The relevant equation to compute the s – maturity simple forward rate i(t, s)’s historical volatility is

var log √ [ (

i (t + Δt, s + Δt) 1 = 𝜎s )] Δt i (t, s)

(25.9b)

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This is the standard deviation of the percentage change2 in the s – maturity simple forward rate over [t, t + Δt]. Multiplication by (1/Δt) annualizes the volatility estimate.

HISTORICAL VOLATILITY To compute the average forward rate volatility as given in expression (25.9a), we need to use some basic ideas from integral calculus. Consider a continuous function y = g(x) as depicted in Figure 25.3, where x is plotted along the horizontal axis and y is plotted along the vertical axis. Calculus tells us that b

the integral ∫ g(x)dx measures the area under this curve between x = a and x = b. a

Recall that one can approximate this area using the set of rectangles lying below the curve (see Figure 25.2). Of course, the area of each rectangle equals its base times its height. The limit of the area covered by these rectangles as the bases go to zero yields the integral of g(x) with respect to x from a to b. We use this simple approximation to compute the average forward rate volatility. The procedure is as follows: (1) collect time series observations of the constant s – maturity forward rates for the entire term structure of maturities s ranging from 0 to time T, (2) compute the variance of the percentage changes in these simple forward rates using Equation 25.9b, (3) add up the variances to get the approximation to the numerator of relationship (25.9a), and finally, (4) divide by the length of the entire time interval T to compute the average. Finally, taking the square root gives the average forward rate volatility. The ratio log[i(t + Δt, s + Δt)/i(t, s)] gives the percentage change in the s — maturity simple forward rate over the time interval. Note that these forward rates correspond to the time intervals [t, t + 𝛿] and [t + Δt, t + Δt + 𝛿], respectively, where 𝛿 is fixed. In our example, 𝛿 corresponds to six months. 2

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CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

As the historical volatility is estimated using actual forward rate realizations that nature “draws,” this estimate is obtained under the actual probability distribution. Using our earlier terminology, we refer to this as “estimation under the probability q.” The next example illustrates this method.

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EXAMPLE 25.4: Computing the Historical Average Forward Rate Volatility ■

Both the maturity of the caplet, time T, and the simple forward rate’s investment interval, 𝛿, are given by the contract’s specifications. We choose T = 1.25 years and 𝛿 = 0.5 years, representing a typical caplet’s parameters.

■

First, one must decide on the frequency (daily, weekly, or monthly) of the data. Then, one must collect the FRA quotes from dealers in the OTC markets for all maturity FRAs from time 0 to time T. Weekly observation intervals are a good choice, balancing microstructure noise versus providing a sufficient number of observations.

■

Table 25.2 reports hypothetical constant maturity forward rates for maturities ranging from time 0 to T = 1.25 years, collected each week for twenty consecutive weeks. Each forward rate applies to a six-month investment period, that is, 𝛿 = 182/365 = 0.4986 year. - The first column is labeled Week n, and it keeps track of the dates when the data points were collected. - The next six columns give the constant s – maturity forward rates. These simple forward rates always correspond to a constant s – years in the future. Each rate is for a six-month investment period. For example, on week 1, the forward rate for maturity time 0 (the spot rate) is 0.1000. The spot rate corresponds to the investment period [0,0.5]. Next, the forward rate for maturity time 0.25 is 0.1050, corresponding to the rate over the investment period [0.25, 0.75], and so forth.

■

There are several ways to compute the sample variance: - There’s the tedious way of hand computing. Although inefficient, it builds mental muscles. When computing by hand, one needs to compute both the mean and variance for each s – maturity forward rate time series: N 1 Mean ∶ 𝜇̂ s = i (n, s) (25.10a) N∑ n =1 N

Variance ∶

𝜎̂ 2s

1 2 = [i(n, s) − 𝜇̂ s ] N−1∑ n =1

(25.10b)

- Alternatively, one can use a programmable calculator, or a spreadsheet, or a statistical package on a computer. We input the data in Table 25.2 in Microsoft Excel and compute the variance in one step. For example, running Excel command VARA over the twenty observations in cells B2 = 0.1000 to B21 = 0.0920 gives VARA(B2:B21) = 0.00004779. We denote this by 𝜎̂ 0s and record this below the three-month spot rate series. Similarly, we compute and record in the table 𝜎̂ 20.25 = 0.00004984, 𝜎̂ 20.5 = 0.00005241, and so on. ■

Once we have these sample variances, it’s easy to compute the average forward rate volatility.

THE INPUTS

635

TABLE 25.2: Computing the Average Forward Rate Volatility Using Weekly Price Data

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Constant Maturity Simple Forward Rate i(n, s) Week n

s=0

s = 0.25

s = 0.5

s = 0.75

s = 1.00

s = 1.25

1

0.1000

0.1050

0.1100

0.1150

0.1200

0.1150

2

0.1020

0.1060

0.1120

0.1160

0.1230

0.1160

3

0.1050

0.1070

0.1140

0.1160

0.1260

0.1190

4

0.1080

0.1090

0.1150

0.1180

0.1270

0.1230

5

0.1090

0.1110

0.1150

0.1210

0.1280

0.1250

6

0.1100

0.1130

0.1180

0.1210

0.1290

0.1280

7

0.1100

0.1150

0.1190

0.1250

0.1300

0.1310

8

0.1110

0.1150

0.1200

0.1240

0.1340

0.1310

9

0.1120

0.1160

0.1210

0.1270

0.1340

0.1350

10

0.1120

0.1170

0.1210

0.1280

0.1350

0.1360

11

0.1110

0.1180

0.1210

0.1270

0.1360

0.1310

12

0.1090

0.1170

0.1200

0.1260

0.1350

0.1280

13

0.1070

0.1130

0.1180

0.1230

0.1290

0.1210

14

0.1030

0.1090

0.1100

0.1200

0.1230

0.1240

15

0.0990

0.1050

0.1100

0.1160

0.1210

0.1210

16

0.0980

0.1030

0.1080

0.1180

0.1200

0.1190

17

0.0960

0.1010

0.1060

0.1090

0.1160

0.1160

18

0.0940

0.0990

0.1010

0.1050

0.1140

0.1110

19

0.0920

0.0970

0.1000

0.1030

0.1110

0.1090

20

0.0920

0.0950

0.0990

0.1010

0.1090

0.1080

Variance

0.00004779

0.00004984

0.00005241

0.00006542

0.00006747

0.00006940

Variance × 0.25

0.00001195

0.00001246

0.00001310

0.00001635

0.00001687

0.00001735

Summed variance × 0.25/ T

0.00007046

Volatility × √52 Tis 1.25 years; 𝛿 is 0.5 years.

0.060530

636

CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

- Multiply each of these variance estimates by 0.25 (the base of the rectangle) and record them in Table 25.2. Summing gives the area under the curve, that is, T

∫ 𝜎2s ds ≈ 0

∑

𝜎̂ 2s × 0.25 = (𝜎̂ 20 + 𝜎̂ 20.25 + … + 𝜎̂ 21.25 ) × 0.25 = 0.00008808

(25.10c)

s

- Dividing (25.10c) by the length of the time interval T gives 𝜈2T =

∑s 𝜎̂ 2s × 0.25 = 0.00007046 T

(25.10d)

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The positive square root is the average forward rate volatility 𝜈T = 0.008394, or 0.8394 percent per week. To get the volatility per year, multiply by the square root of 52, to get 0.060530 or 6.0530 percent. This represents the average volatility (per year) of the constant maturity forward rates over the caplet’s life.

In computing the historical average forward rate volatility, a number of issues still remain. First, how many observations should you use? Theoretically speaking, more observations will give a better estimate, but long time series data may not be “stationary.” A good rule of thumb is to use at least 50 observations so that the central limit theorem applies. Second, how long should the time interval be between observation dates? If observations are collected at short intervals, then market frictions and market microstructure issues can significantly impact the estimation. Daily or weekly intervals have fewer of these problems. Finally, what should be the time interval between the maturities of the forward rates? The smaller the time interval, the better that the sum of the rectangles approximates the area under the curve. In practice, quarterly time intervals between the different maturity forward rates, as in Table 25.2, yield a reasonable approximation.

IMPLIED VOLATILITY Just as for the BSM formula, one can use market prices of caps, as portfolios of caplets, to find that average forward rate volatility 𝜈T that best matches market prices. This is calibration. The procedure yields an implied volatility. In practice, just as in the BSM model, implied volatilities for caplets differ from historical volatilities, and when graphed across strikes and maturities, they exhibit a smile or sneer (see Jarrow et al. 2007; Rebonato 2002). This implies, as in the case of the BSM formula, that the HJM libor model is formally rejected by the data. Nonetheless, the model can still be used with implicit volatilities but, in this case, only as a statistical model for pricing caplets. When used as a statistical model, its purpose must be restricted to pricing caplets (and floorlets) only, and its use as a hedging tool is no longer appropriate. This is because the theoretical model has been rejected (see Chapter 20 for a detailed discussion of these issues). The comments and criticisms previously mentioned with respect to the BSM formula apply to the HJM libor model as well. As with options in other markets, many dealers express their quotes in terms of volatilities, which they refer to as “vols.” The other inputs being known, with the HJM libor model you can determine the caplet price.

UNDERSTANDING THE HJM LIBOR MODEL

25.7

Understanding the HJM Libor Model

Recall martingales, pseudo-probabilities, and risk-neutral valuation from equity option pricing. We introduced them in part III of the book to provide an enriched understanding of the option pricing methodology. We do the same here. In addition, these concepts also enable us to provide a quick sketch of the proof for the caplet pricing formula in Result 25.1.

Forward Rates and Martingales As discussed in the context of the binomial interest rate option pricing model of Chapter 23, it can be shown (see the appendix to this chapter)3 that the no-arbitrage assumption implies the existence of pseudo-probabilities 𝜋T+𝛿 (z) > 0 such that i (0, T) = E𝜋T+𝛿 [i (T, T)]

(25.11)

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where i(0,T) is today’s (time 0) simple forward rate for the time interval [T,T + 𝛿], T is the caplet’s expiration date, 𝛿 is the time period over which the interest is applied, and E𝜋T+𝛿 [.] is expectation based on these pseudo-probabilities 𝜋T+𝛿 (z). Here the pseudo-probabilities depend on the maturity of the forward rate. Equation 25.11 states that the current forward rate equals its expected time T value, when using the pseudo-probabilities. Interestingly, this shows that the unbiased expectations hypothesis (which was introduced in Extension 21.2) holds under the pseudo-probabilities. Because the pseudo-probabilities contain an adjustment for risk (see below), this also shows that the unbiased expectations hypothesis will not hold under the actual probabilities.

Risk-Neutral Valuation To see how a caplet is priced by risk-neutral valuation, let us first note that as shown in the appendix to this chapter the caplet’s value can be written as c (0) = B (0, T + 𝛿) E𝜋T+𝛿 {max [i (T, T) − k, 0] ⋅ 𝛿} × LN

(25.12)

where E𝜋T+𝛿 {.} is expectation based on the pseudo-probabilities 𝜋T+𝛿 (z). In this equation, the caplet’s payment date is time (T + 𝛿). This expression states that the caplet’s time 0 value is equal to its discounted expected payoff, using the pseudo-probabilities. As such, it is analogous to the riskneutral valuation procedure used with respect to the BSM model. Equation 25.12 provides us with a convenient method for proving Equation 25.7a of Result 25.1. As noted earlier, under the pseudo-probabilities, the forward rate i(T,T) has a lognormal distribution with mean zero and variance 𝜈2T . A direct 3

It can be shown that the lognormally distributed simple forward rates imply a stochastic process for the corresponding continuously compounded forward rates. Condition (25.11) can then be used to show that the HJM arbitrage-free drift restriction holds for the continuously compounded forward rate process, proving that the implied term structure evolution is arbitrage-free.

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CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

evaluation of the expectation in Equation 25.12 using this lognormal distribution gives the HJM libor formula. The proof is identical to that given for the BSM model, with the exception that only the inputs (and their interpretations) have changed.

Actual versus Pseudo-probabilities The difference between the actual and pseudo-probabilities represents an adjustment for risk, just as for the BSM model. To see this, we start with assumption A5. Under assumption A5, the simple forward rate i(0,T) at time 0 has a lognormal distribution. We can write this as T

log

i (T, T) 1 = ∫ 𝜇s ds − 𝜈2T T + 𝜈T √T ⋅ z ( i (0, T) ) 2 0

(25.13)

where z is a standard normal random variable with mean 0, variance 1, with 1 2 − z

probability q(z) = e 2 /√2𝜋. The Equation 25.13 holds at any time t between time 0 and the expiration date T. These q(z)s are the actual probabilities. As with the BSM model, it can be shown (see the appendix to this chapter) that the actual probabilities and the pseudoprobabilities must agree on zero probability events, that is, 𝜋T+𝛿 (z) > 0 if and only if q (z) > 0

(25.14)

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Otherwise, traders will not agree on what constitutes an arbitrage opportunity. Second, this implies that the actual and pseudo-probabilities are related by: 𝜋T+𝛿 (z) = 𝜑T+𝛿 (z) q (z) where −

𝜑T+𝛿 (z) = e

𝜃T+𝛿 = −

𝜃2T + 𝛿 2

+ 𝜃 T +𝛿 z

∫0T

and

𝜇s ds

( 𝜈 √T ) T

(25.15a)

(25.15b) (25.15c)

The quantity 𝜑T + 𝛿 (z) is an adjustment for risk. The justification for this statement is the same as that given for the BSM formula in Chapter 19. This risk adjustment in the pseudo-probabilities is the reason that the risk-neutral valuation procedure works in our world with risk-averse traders!

25.8

Caps and Floors

The HJM libor model for pricing caplets is all we need! One can use it to price a cap (which is a collection of caplets), a floorlet (via caplet–floorlet parity), and a floor (which is a collection of floorlets). Let’s now see how to price these interest rate derivatives.

CAPS AND FLOORS

Caps A cap is a financial security with a notional amount LN , a strike rate k, an expiration date T, and a “payment” interval 𝛿. The cap buyer receives the payment max [i (t − 𝛿, t − 𝛿) − k, 0] × 𝛿 × LN

(25.16)

at the payment dates t = 𝛿, 2𝛿, ..., (T + 𝛿). For example, the first payment happens at time 𝛿, which is computed using the simple forward rate i(0, 0) that applies over the time interval [0, 𝛿]. Recall that this simple forward rate effective immediately is also the spot rate. The second payment happens at time 2𝛿 and is based on the forward rate i(𝛿, 𝛿) known at time 𝛿, and so forth. Hence, a cap is a portfolio of caplets, with each caplet having the same strike rate k as the cap, and each with a different maturity date corresponding to the payment dates. We can find this cap’s value by valuing each individual caplet and adding the resulting values.

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Floorlets A floorlet is a European put option on the spot rate of interest. As with a caplet, the payment is received 𝛿 years (a fraction of a year) after the floorlet’s stated maturity at time T. The payment equals the notational times the simple interest rate [k − i(T, T)]𝛿 adjusted for the time period 𝛿 over which the interest applies. To dress this up in more formal terms, consider a floorlet with notional principal LN , strike rate k, and expiration date T, where the time period over which interest is computed is 𝛿 years and the payment date is (T + 𝛿). The floorlet holder’s payoff on the payment date is pT+𝛿 = max [k − i (T, T) , 0] × 𝛿 × LN

(25.17)

where the simple forward rate i(T, T) is the interest rate determined at time T that applies over the time period [T, T + 𝛿]. To obtain the value of this payoff on the floorlet’s maturity date, multiply the preceding payoff by B(T, T + 𝛿), which is the time T price of a zero-coupon bond that pays a dollar at time (T + 𝛿). Once you have priced a caplet, it’s easy to value a floorlet by using our old friend caplet–floorlet parity.

Caplet–Floorlet Parity Just as the price of otherwise identical European calls and puts on the same underlying are related by put–call parity, so are the prices of caplets and floorlets. The following argument is a generalization of that given to derive caplet floorlet parity in the HJM binomial model. We start with a mathematical identity: i (T, T) − k = max [i (T, T) − k, 0] − max [k − i (T, T) , 0]

(25.18a)

You can easily check that this identity is true by inputting values for the simple forward rate i(T, T) that are greater than or less than the common strike rate k. Notice that the payoffs on the right side are those at time (T + 𝛿) for the caplet and floorlet, respectively. The left side represents the payments to an FRA with maturity

639

640

CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

time (T + 𝛿). Next, we apply the risk-neutral valuation procedure: take expectations E𝜋T+𝛿 {⋅} using the pseudo-probabilities 𝜋T + 𝛿 (z) and then discount the payoffs by multiplying by the zero-coupon bond’s price that matures at time (T + 𝛿). This gives B (0, T + 𝛿) E𝜋T+𝛿 [i (T, T) − k] × 𝛿 × L = B (0, T + 𝛿) E𝜋T+𝛿 {max [i (T, T) − k, 0]} × 𝛿 × LN − B (0, T + 𝛿) E𝜋T+𝛿 {max [k − i (T, T) , 0]} × 𝛿 × LN

(25.18b)

where we have also multiplied the equations on both sides by 𝛿 × LN , the time period over which interest is computed times the notional. Because the expectation of i(T, T) is today’s simple forward rate, the expected value of the constant k is itself and the discounted expected value of the caplet’s and floorlet’s payoffs are today’s prices, we get the final result (see Result 25.2).

RESULT 25.2 Caplet–Floorlet Parity c − p = B (0, T + 𝛿) [i (0, T) − k] × 𝛿 × LN

(25.19)

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where c and p are today’s (time 0) prices of caplets and floorlets that have strike rate k and expiration date T, respectively, B(0, T + 𝛿) is today’s price of a zero-coupon bond that pays $1 at time (T + 𝛿), i(0, T) is the simple forward rate that applies to the period [T, T + 𝛿], and LN is the notional principal.

Example 25.5 uses caplet–floorlet parity to price a floorlet.

EXAMPLE 25.5: Pricing a Floorlet Using the HJM Libor Model ■

Consider the CFP data. - Today is time 0, the floorlet’s expiration date T is one year, interest applies over six months, a 𝛿 = 182/365 = 0.4986 year period, and the payoff date is (T + 𝛿). - The notional principal LN = $100 million and the strike rate k = 4 percent per year. - The simple forward rate applicable over 𝛿 is i(0,T) = i(0,1) = 0.0421. - The average forward rate volatility 𝜈T is 0.15, or 15 percent per year. - A zero-coupon bond maturing after 1.4986 years is worth B(0, T + 𝛿) = B(0, 1.4986) = $0.95 today.

■

Example 25.3 gave the caplet’s price as c = B(0, T + 𝛿) [i (0, T) N(d1 ) − kN(d2 )] × 𝛿 × LN = $172, 842.45

USING THE HJM LIBOR MODEL

■

Using the caplet–floorlet parity relation (25.19), the floorlet’s price is p = c − B (0, T + 𝛿) [i (0, T) − k] × 𝛿 × LN = 172, 842.45 − 0.95 × (0.0421 − 0.04) × 0.4986 × 100 million = $73, 116.42

Floors A floor is a financial security on a notional amount LN with a strike rate k, a maturity date T, and a “payment” interval 𝛿. A floor holder receives payments max [k − i (t − 𝛿, t − 𝛿) , 0] × 𝛿 × LN

(25.20)

at times 𝛿, 2𝛿, . . . , T, (T + 𝛿). Thus a floor is a portfolio of floorlets, each with the same strike rate k as the floor but with different maturity dates corresponding to the payment dates. To value a floor, price the constituent floorlets and just add them up!

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25.9

Using the HJM Libor Model

Using a derivative pricing model really means “hedging.” After all, a perfect hedge is the basis for derivatives pricing, and the holy grail of option pricing is the determination of the option’s delta. Let’s see how we can hedge in the HJM libor model. Stepping back for one moment, hedging interest rate risk in an HJM model is very flexible because every traded security can be used to hedge anything else. However, because of the special structure of the HJM libor model, this section confines itself to a discussion of delta and gamma hedging for caplets and floorlets. There are several reasons for this. 1. Caplets and floorlets have closed-form solutions in the HJM libor model. This enables one to easily compute partial derivatives yielding exact formulas for delta and gamma hedges. Unfortunately, closed form solutions are not available for other interest rate derivatives. 2. Our discussion of the dangers in using vega hedging in a model where the volatility is nonrandom and not priced within the model applies to the HJM libor model. This is confounded by the fact that the HJM libor model is rejected by historical data, exactly analogous to the BSM model (see Chapter 20). 3. The delta and gamma hedging techniques discussed here do not extend to more complex interest rate derivatives like American options and futures contracts, which do not have closed-form solutions. They must be hedged using numerical methods in a more complex HJM model, examples of which were discussed in

641

642

CHAPTER 25: THE HEATH–JARROW–MORTON LIBOR MODEL

chapter 24 in the context of a two-period binomial model. This is because the continuously compounded forward rates in an HJM libor model, which are needed to price interest rate derivatives that are not just simple extensions of caplets and floorlets, do not follow a simple probability distribution.

The Greeks Hedging the Greeks is a risk management technique that uses the HJM libor model and a Taylor series expansion to hedge the interest rate risk of a cap or floor position. The approach is similar to the one discussed in Chapter 20 in the context of the BSM model. Before introducing Greek hedging, let us first compute the deltas and gammas needed in the hedges. ■

Taking the partial derivative of the caplet pricing formula in Result 25.1 with respect to i(0,T) ≡ i, it can be shown that the caplet’s delta is DeltaC ≡

𝜕c 𝜕i

= −

𝛿 c + B (0, T + 𝛿) N (d1 ) 𝛿LN [1 + i (0, T) 𝛿]

(25.21a)

This makes sense because all else being equal, an increase in the simple forward rate raises a caplet’s value.

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■

We can rewrite the definition of the simple forward rate given in Equation 25.3a as B (0, T) (25.21b) B (0, T + 𝛿) = 1+i×𝛿 Taking the partial derivative of this equation with respect to i(0,T) gives 𝜕B (0, T + 𝛿) B (0, T + 𝛿) 𝛿 =− 0. Let us consider some VaR computations to help us understand its application and significance.

FIGURE 26.3: Value-at-Risk Probability density

α percent probability

–Varα

0

∆At – ∆Lt

VALUE-AT-RISK AND SCENARIO ANALYSIS

To understand how to use VaR𝛼 in risk management decisions, let us suppose that 𝛼 = 0.01 is the insolvency probability that the firm and the regulators feel is appropriate. Given the VaR𝛼 , the firm’s management can consider if it is properly capitalized for its insolvency risk by comparing VaR𝛼 to the firm’s existing equity capital position Et .

EXAMPLE 26.1: Computing Value-at-Risk ■

To compute VaR, one views the firm as a portfolio of “securities,” where the securities are the firm’s assets and liabilities. The assets have a positive portfolio weight, while the liabilities have a negative weight. Of course, the portfolio weights must sum to one.

■

Consider a firm which has an asset and liability portfolio consisting of three companies (fictitious names) Aztec, Inca, and Mayan corporations, with portfolio weights as given in Table 26.2A. Furthermore, let the daily returns on these three companies be normally distributed with means, volatilities, and correlations as given in Tables 26.2A and 26.2B. This characterizes the random evolution of the primary securities underlying the firm’s assets and liability portfolio.

■

To compute a daily VaR based on this normal distribution, we need to first compute the expected daily return and volatility of the firm’s asset and liability portfolio. The daily expected return of this portfolio is 3 E (rp ) =

∑

wi E (ri )

i =1

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= (3/4) × (0.001) + (3/4) × (0.0005) − (1/2) × (0.001) = 0.000625 The daily return variance of this portfolio is 3

𝜎2p

=

∑

i,j =1

w2i,j covariance (ri , rj )

= (3/4)2 (0.0134164)2 + (3/4)2 (0.0089443)2 + (−1/2)2 (0.0178885)2 +2 (3/4) (3/4) (0.0134164) (0.0089443) (0.5) +2 (3/4) (−1/2) (0.0134164) (0.0178885) (0.9) +2 (3/4) (−1/2) (0.0089443) (0.0178885) (0.5) = (0.0001012 + 0.000045 + 0.00008) + (0.0000675 − 0.000162 − 0.00006) = 0.0002262 − 0.0001545 = 0.0000717 where we have used the well-known result that covariance (ri , rj ) = 𝜎i × 𝜎j × correlation (ri , rj ), where i and j denote the companies, which are labeled 1, 2, and 3, respectively.

663

664

CHAPTER 26: RISK MANAGEMENT MODELS

■

The daily volatility is the square root of the variance: 𝜎p = +√0.0000717 = 0.0084676

TABLE 26.2A: Expected Return, Volatility, and Portfolio Weights Aztec Corp. (Company 1)

Inca Corp. (Company 2)

Mayan Corp. (Company 3)

0.001

0.0005

0.001

0.0134164

0.0089443

0.0178885

3/4

3/4

-1/2

Expected daily return Daily volatility of the returns Portfolio weights

TABLE 26.2B: Correlation of Expected Returns Aztec Corp.

Inca Corp.

Mayan Corp.

1

0.5

0.9

Inca Corp.

0.5

1

0.5

Mayan Corp.

0.9

0.5

1

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Aztec Corp.

■

Daily

Third, we can now compute the daily VaR at the 5 and 1 percent levels. Recall from statistics that for a normal probability distribution, 95 percent of the values lie within 1.65 standard deviations of the mean. - The 5 percent left tail is 1.65 × 0.0084676 = 0.0139715 units away from the mean. - Hence the 95 percent confidence level or −VaR0.05 is 0.000625 − 0.0139715 = − 0.0133465. - This means that there is only a 5 percent chance that the firm will lose more than 1.33 percent of its value on any given trading day.

■

As 99 percent of the values lie within 2.33 standard deviations of the mean, we get the following: - The 1 percent left tail is 2.33(0.0084676) = 0.0197295 units away from the mean. - The 99 percent confidence level or −VaR0.01 is 0.000625 − 0.0197295 = − 0.0191045. - Thus there is only a 1 percent chance that the firm will lose more than 1.91 percent of its value on any given trading day.

■

If VaR𝛼 < Et , then the firm’s insolvency probability is less than the 𝛼 probability, and the firm is properly capitalized. Indeed, if a loss equal to VaR𝛼 occurs, then there is enough equity capital to buffer this loss. The firm does not become insolvent.

VALUE-AT-RISK AND SCENARIO ANALYSIS

■

If VaR𝛼 > Et , then the firm’s insolvency probability is greater than 𝛼, because there is not enough equity capital to cover the VaR𝛼 losses. The firm needs either to (1) add more equity capital to its balance sheet or (2) restructure the risk of its assets and liabilities to reduce the loss probability. For example, (1) the firm can issue new equity or cut dividend payments and (2) it could sell assets and use these funds to reduce its liabilities.

If it is used as the sole measure to quantify a firm’s risk, VaR𝛼 has a number of failings. First, it does not consider the magnitude of the losses beyond the 𝛼 level. For example, if the VaR𝛼 is $100 million, a 1 percent loss of an additional $150 million beyond the VaR𝛼 is treated equally in the VaR𝛼 computation as an additional $500 million loss beyond VaR𝛼 . Clearly these different losses should not be treated as of equal risk to the firm. Second, in particular situation, VaR𝛼 penalizes diversification. Traditional portfolio theory views diversification of an asset and liability portfolio as beneficial because it reduces risk. The fact that VaR𝛼 can increase instead of decrease when a portfolio becomes more diversified is problematic. This possibility is illustrated in the next example.

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EXAMPLE 26.2: VaR𝛼 Penalizes Diversification ■

Strangely, VaR𝛼 can increase when a portfolio becomes more diversified! This is counter to the intuition that you developed taking a modern investment course where diversification reduces risk. This implies, of course, that VaR𝛼 is not always a good measure of a firm’s risk.

■

Suppose that there are two loans A and B that can be used to constitute a firm’s assets. Let A and B have the loss distributions given in Table 26.3. Loans A and B are similar in that both have no loss with probability 0.9991 and a $1 loss with probability 0.0009. We assume that the probabilities of default are statistically independent for loans A and B.

TABLE 26.3 Loss

Prob(loss A)

Prob(loss B)

Prob(loss [A+ B]/2)

$0

0.9991

0.9991

0.99820081

$0.5

0

0

0.00179838

$1

0.0009

0.0009

0.000000081

■

We compare two hypothetical firms. One firm invests a dollar in loan A. The second firm invests $0.50 equally in both loans A and B. The second firm, of course, has the more diversified loan portfolio because it is spreading its $1 investment across two loans whose losses are statistically independent.

■

The second firm’s return is given in the third column. With probability (0.0009)(0.0009) = 0.000000081, both loans A and B will default, giving a $1 (= $0.5 + $0.5) loss to the second firm. With probability 2(0.0009)(0.9991), either loan A or loan B will default, but not both, giving a $0.50

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(= $0.5 + $0) loss to the second firm. With probability (0.9991)(0.9991), neither loan will default, giving a $0 loss to the second firm. ■

Let us choose 𝛼 = 0.001 for our VaR computation. The nondiversified firm consisting of only loan A has VaR0.001 (A) = $0 because no losses occur with probability greater than 0.001. But the diversified firm has VaR0.001 ([A + B]/2) = $0.50 because with probability 0.00179838, a loss of $0.5 happens. The diversified firm’s VaR is larger! Hence the diversified portfolio would require more capital based on this measure alone, which is an incorrect decision.

Because diversification reduces risk, this last failing is a serious flaw in using VaR𝛼 as a risk measure. Used in isolation for the determination of equity capital, VaR𝛼 is a dangerous statistic because of the two failings just discussed. However, used in conjunction with other risk measures, it is a useful tool in the risk manager’s toolbox.

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Scenario Analysis During the financial crisis of 2007, the Federal Reserve Board required banks receiving funds under the Troubled Asset Relief Program to compute an alternative risk measure to decide if they could repay their TARP funds.3 This alternative measure is based on scenario analysis, another useful technique in one’s risk management toolbox. Scenario analysis (which also goes by the popular name stress testing) starts with a selection of a particular set of scenarios that the firm wants to protect against. These scenarios correspond to various states of the economy represented by paths of the state variables ΔXt over the time period [t, t + Δt]. For example, if Xt represents a collection of macroeconomic variables (such as inflation, gross domestic product, unemployment, and interest rates), then a set of scenarios might be that interest rates increase by 1 percent, unemployment increases by 5 percent, inflation increases by 5 percent, and GDP falls by 2 percent. For these scenarios, the firm wants to compute the losses realized on its assets and liabilities to see if it has sufficient equity capital to avoid insolvency. Using our notation, the firm picks a particular ΔXt and computes ΔEt (Xt ) = ΔAt (Xt ) − ΔLt (Xt ) Given the formulas for the values of the assets and liabilities as a function of the state variables, one then computes the change in the firm’s equity capital, and if it is negative and exceeds the current equity capital position, one knows that this scenario would cause the firm to become insolvent.

3

“Pay up! The U.S. Federal Reserve Asks Banks to Submit TARP Repayment Plans,” New York Daily News, November 24, 2009.

VALUE-AT-RISK AND SCENARIO ANALYSIS

667

EXTENSION 26.1: Risk Measures Value-at-risk is a useful risk measure, but as noted in the text, it has some flaws. As might be expected, there are other risk measures that could be used either in conjunction with or instead of VaR𝛼 . The study of risk measures for use in the determination of equity capital began with the paper by Artzner et al. (1999). They defined coherent risk measures. To understand what these are, we need to step back and begin with the definition of a risk measure. A risk measure is a positive real valued function that quantifies the “risk” to changes in a firm’s equity capital—losses—over some time period. The loss to a firm’s equity capital is, of course, a random variable. By this definition, there are an unlimited number of risk measures. For example, the maximum loss, the expected loss, the standard deviation of the loss, VaR𝛼 for an arbitrary 𝛼, and the present value of the loss are all risk measures. This leads to the question, how does one decide which risk measures are useful? The answer is that to be of any use, a risk measure needs to possess some desirable properties or axioms that make it a “good” risk measure. Artzner et al. defined four such axioms. To state these axioms, we need some additional notation. Let 𝜌( . ) be a risk measure and let ΔE be the change in a firm’s equity over a fixed time period. Let r be the return on a riskless asset (a money market account) over this same time period. Now we can state the four axioms: (Translation invariance) 𝜌 (ΔE + α) = 𝜌 (ΔE) − α (Subadditivity) 𝜌 (ΔE1 + ΔE2 ) ≤ 𝜌 (ΔE1 ) + 𝜌 (DE2 ) Copyright © 2018. World Scientific Publishing Company Pte. Limited. All rights reserved.

(Positive homogeneity) For λ ≥ 0, 𝜌 (λΔE) = λ𝜌 (ΔE) (Monotonicity) For ΔE1 ≤ ΔE2 , 𝜌 (ΔE2 ) ≤ 𝜌 (ΔE1 ) The translation invariance axiom states that if one adds 𝛼 units of equity capital to the firm and invests it in a riskless asset, the risk measure declines by exactly 𝛼 units. Subadditivity implies that a diversified portfolio (combining two firms’ equity capital) is no more risky than the sum of each separate firm’s capital positions. Positive homogeneity states that if you expand a firm’s assets and liabilities proportionately, the risk increases proportionately as well. Last, monotonicity states that if changes in a firm’s equity capital always exceed another firm’s (both plus and minus), then the firm with the larger changes is more risky. Any risk measure that satisfies these four axioms is called a coherent risk measure. As one might expect, there are many coherent risk measures and many risk measures are not coherent. For example, VaR𝛼 is not a coherent risk measure because as shown in a previous example in the text, it violates the subadditivity axiom. Perhaps surprising the premium paid for an insurance policy to insure against the firm’s default is not a coherent risk measure either because it violates the translation axiom. An example of a coherent risk measure is the expected value of ΔE. A more sophisticated coherent risk measure ̃ ̃ ≤ −VaR𝛼 }, where E{.|ΔE ≤ −VaR𝛼 } denotes a conditional is the expected shortfall, defined as E{ΔE|ΔE expectation given that losses in equity values exceed VaR𝛼 . The proof that expected shortfall is a coherent measure can be found in Follmer and Schied (2004). Whether coherent risk measures are the proper risk measures depends on whether the four axioms are deemed reasonable, and there is considerable disagreement on this point. For example, Jarrow (2002a) does not like the

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translation invariance axiom precisely because it excludes the premium paid for an insurance policy to insure against the firm’s default as a coherent risk measure. To rectify the situation, Jarrow proposes an alternative set of axioms that generates what he calls insurance risk measures. Other sets of axioms are also possible, and other risk measures can be characterized in this manner. The study of risk measures is an exciting area of current research (for a good summary of the different axiom selections, see Follmer and Schied 2004).

In this circumstance, if it is believed that this scenario is likely, the firm can either (1) add more equity capital to its balance sheet or (2) restructure the risk of its assets and liabilities to reduce the loss probability. Given that the loss is scenario specific, it might also be possible to hedge this loss using derivatives, as discussed in the previous chapters. The problem with scenario analysis is that it does not assign probabilities to the various scenarios selected. Hence the firm’s management does not know the probability that this scenario will occur. Of course, this failing can be overcome by selecting the scenarios to be considered in conjunction with the probability distribution for ΔXt . Unfortunately, this procedure and related computations is left for more advanced courses.

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26.5

The Four Risks

The Basel Committee’s Risk Management Guidelines for Derivatives (July 1994) identified the following risks in connection with an institution’s derivative activities.4 In contrast to the Basel Committee, we include legal risk in operational risk. Otherwise, the two sets of definitions are identical. ■

Market risk is the risk to a firm resulting from adverse movements in market prices.

■

Credit risk is the risk that a counterparty will default on an obligation.

■

Liquidity risk is the risk that a firm may not be able to liquidate an asset or liability position without incurring losses due to a quantity impact on the market price.

■

Operational risk is the risk that law suits, human error, operation failures, and inadequate controls will result in losses.

This section discusses each of these risks and how to model them using the derivatives pricing technology.

Market Risk The pricing and hedging of market risk using derivatives has been the emphasis of this book. The market risks considered in Chapters 1–25 included changes to stock prices, commodity prices, foreign currency exchange rates, and interest rates. Our models were generated under a set of assumptions that excluded credit risk (by 4

See www.bis.org.

THE FOUR RISKS

assumption A2), liquidity risk (by assumptions A1 and A3), and operational risk (obviously!). To include these additional risks, we need to relax the assumptions and consider more general models. The remainder of this section discusses these generalizations.

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Credit Risk Credit risk is the risk that a counterparty will default on a contract. This is an important consideration in trading activity because all counterparties may default. An example serves to illustrate this risk. On June 26, 1974, the West German bank Bankhaus Herstatt, which had received Deutschemark payments, was forced into liquidation by West German regulators before it could make corresponding dollar payments in New York (which had a six-hour time difference), causing losses for its counterparties. This incident alerted the market about cross-currency settlement risk, also known as Herstatt risk (which Reuters Financial Glossary defines as “the risk that when dealing with an overseas client a bank may not be able to recover its funds if the counter party defaults on its payment obligation”5 ). This event led to the development of a continuously linked settlement system for foreign exchange payments. There are two approaches to modeling credit risk: structural and reducedform models. Structural models are sometimes called contingent claims models, as originated by Merton (1974). Jarrow and Turnbull (1992, 1995) originated reduced-form models to overcome a restrictive assumption contained in structural models. Structural models assume that all of the firm’s assets trade in frictionless and competitive markets and, therefore, their prices are observable. This assumption is usually violated in practice. Most firms’ assets are very illiquid and often do not trade, for example, a firm’s home office building or manufacturing plant, including the manufacturing equipment. In contrast, reduced-form models replace this assumption with a weaker restriction. Reduced-form models only assume that a subset of the firm’s liabilities (debt) trade. The subset of the firm’s liabilities that must trade is the one for which pricing and hedging is being considered. This small difference, however, leads to a big change in valuations and risk management. From an economic perspective, structural models are best applied to internal management decision making, in which, even though the assets do not trade in frictionless and competitive markets, the management can “observe” their prices. In this context, structural models aid economic intuition. In contrast, reduced-form models are best applied to pricing, hedging, and risk management decisions that involve the trading of credit risky securities in financial markets. As such, they are the preferred modeling approach to risk management in our current market environment. They are easily estimated, and empirical studies show that they fit market prices well (for a review, see Jarrow 2009).

5

http://glossary.reuters.com/index.php/Herstatt_Risk.

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TABLE 26.4: A Firm’s Time t Balance Sheet Assets At

Liabilities Lt

Zero-coupon bond

Maturity T, Face value K

Equity Et

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Structural Models To understand the intuition behind the structural models, it is best to start with Merton’s (1974) original formulation. We consider a simple firm with the liability structure given in Table 26.4. As before, this firm has assets with time t value At against which it has issued debt. The debt consists of a single zero-coupon bond with face value K and maturity date T. We let Lt be the time t value of the firm’s debt and Et be the time t value of the firm’s equity. This is the simplest liability structure a firm can possess. This firm is too simple for this model to be used in practice, but it is just right for us to understand the economics. Merton’s structural model is a transformation of the BSM model of chapter 19. The structural model uses assumptions A1–A8 from the BSM model, with two modifications. First, assumption A2, no credit risk, is removed because the purpose of the structural model is to understand when the firm will default on its debt. Second, the firm’s asset value replaces the stock price in assumptions A7 and A8. In particular, the structural model assumes that the firm’s assets trade continuously in time and that the asset value process follows a lognormal distribution. Assumption A6, no interest rate uncertainty, is retained. We let the continuously compounded riskless spot rate of interest per year be denoted r. Hence a default-free zero-coupon bond paying a dollar at time T has time t value B (t, T) = e−r(T−t) In this firm, the time T value of the firm’s debt and equity is completely determined by the debt’s covenants. At the debt’s maturity, the equity holders decide whether to pay off the debt. They will do so if and only if the value of the firm’s assets exceeds the debt’s face value. Otherwise, default occurs. Hence the time T value of the equity can be written as ET = max [AT − K, 0] (26.5) We see that the equity’s value is equivalent to the payoff from a European call option on the firm’s value, with maturity date T and strike price K. In essence, the debt holders, and not the equity holders, own the firm. The equity holders have retained the option to buy the firm back from the debt holders at time T for K dollars. They will do so at time T if the value of the assets exceeds this purchase price. Hence we can write the value of the equity at time t to be equal to Et = c (At , K, T)

(26.6)

THE FOUR RISKS

where c(a, b, T) represents the value of a European call option at time t on an asset with current price a, strike price b, and maturity date T. Using the accounting identity relating the value of the firm’s equity to its assets and liabilities, we see that the debt’s time T value is equal to the value of the assets less the value of the equity, that is, LT = AT − ET = AT − max [AT − K, 0]

(26.7)

Simple algebra shows that we can rewrite the debt’s time T value as LT = min [K, AT ] = K − max [K − AT , 0]

(26.8)

The debt’s value at maturity is seen to be equal to the smaller of the face value of the debt or the firm’s asset value. Alternatively, the debt’s time T payoff is equal to the face value of the debt less the payoff to a European put option on the firm’s value, with maturity date T and strike price K. The time t value is therefore

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Lt = Ke−r(T−t) − p (At , K, T)

(26.9)

where p(a, b, T) represents the value of a European put option at time t on an asset with current price a, strike price b, and maturity date T. This shows that the firm’s risky debt is valued less than a similar maturity default-free zero-coupon bond (Ke−r(T−t) ). The decrease in value is equal to the put option’s value in expression (26.9). Using the lognormal distribution assumption, we can give an explicit formula for the debt’s time t value: Lt = At N (−d1 ) + Ke−r(T−t) N (d2 )

(26.10)

where N( . ) is the cumulative standard normal distribution function which has mean 0 and standard deviation 1, log d1 =

At 𝜎2 (T − t) + r+ (k) ( 2) 𝜎√T − t

d2 = d1 − 𝜎√T − t, and 𝜎 is the asset return’s volatility per year. This expression has a nice economic interpretation. The debt’s value is equal to the firm’s value multiplied by N(−d1 ) plus the risk-adjusted probability of no default times the discounted face value of the debt. The two terms represent the possible payoff scenarios for the firm’s debt, depending on if the firm defaults (the first term) or does not (the second term). It is easy to show that as the firm’s asset volatility increases, the value of the firm’s debt decreases. Since the firm’s asset volatility 𝜎 is under the control of management, the decreasing value of the debt as 𝜎 increases leads to the shareholder/debt holder conflict

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(see Ross et al. 2004, p. 640). Understanding the shareholder/debt holder conflict is just one use of the structural model. Another is that it can be used to provide insights into the firm’s investment/capital budgeting decisions. This is the theory of real options (see Extension 26.2). The structural model can be extended to consider more complex liabilities issued by the firm. However, this extension becomes quite difficult to solve analytically because the entire liability structure of the firm needs to be specified and all of the liability values solved simultaneously. In this regard, numerical procedures need to be employed (for some of these extensions, see Jones et al. 1984; Leland 1994).

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EXTENSION 26.2: Real Options Option pricing theory can be used to understand how a firm should make its investment decisions. In the past, this field was called capital budgeting. Today it is called real options, the title of this extension. This is because embedded in most of the firm’s investment decisions are options, which were often ignored in classical capital budgeting. Real options theory usually proceeds under the same assumptions as the structural model. The model uses assumptions A1–A8 from the BSM model, with the same two modifications: first, assumption A2, no credit risk, is removed; second, the firm’s asset value replaces the stock price in assumptions A7 and A8. In particular, it is assumed that a particular subset of the firm’s assets trades continuously in time and that this asset’s value process follows a lognormal distribution. Assumption A6, no interest rate uncertainty, is retained. We let the riskless spot rate of interest per year be denoted r. Of course, just as in the structural model, many of these assumptions can be relaxed, as a result increasing analytical and computational complexity. We explain real options theory using an example. Let us consider a gold mining company that is thinking about purchasing a new gold mine. The purchase price for the mine is P0 dollars. The management has studied the cost of extracting the gold and believes it can extract one production unit (one hundred tons) for a cost of U0 dollars. Gold trades, and it has a market price of At dollars per unit at time t. It takes T time units to extract the gold from the mine. Last, the management also realizes that after the initial unit of gold is extracted from the mine at time T, it can invest an additional K dollars to extract another k units of gold, where 0< k < 1. For simplicity (although this is not necessary for the analysis), we assume that the additional k units each with a present value of AT at time T. This extra opportunity to invest at time T is an embedded real option that the management wants to include in its decision. This embedded option is a European call option on gold with a strike price of K and a maturity date time T. Hence its payoff is CT = max [kAT − K, 0] (1) Under the no-arbitrage and complete market assumption, we know that we can use risk-neutral valuation to determine the present value of the gold within the gold mine. The present value is the sum of the initial unit of gold plus the embedded option, that is, Present value = E𝜋 (AT ) e−rT − U0 + E𝜋 (max [kAT − K, 0]) e−rT

(2)

THE FOUR RISKS

673

Using the fact the price of gold is a martingale under the risk-neutral probabilities, and using some simple algebra on the value of the embedded call option, we can rewrite this as Present value = A0 − U0 + kc0 (A0 , K/k, T)

(3)

where c0 (a, b, T) is the time 0 value of a European call option on an asset with price a, strike price b, and maturity date T. Using the lognormal assumption for the evolution of the gold’s price, we can determine the value of the project to be Present value = A0 − U0 + kA0 N (d1 ) − Ke−rT N (d2 ) (4) where N( . ) is the cumulative standard normal distribution function, which has mean 0 and standard deviation 1, log d1 =

A0 𝜎2 + r+ T ( K/k ) ( 2) 𝜎√T

d2 = d1 − 𝜎√T, and 𝜎 is the price of gold’s return volatility per year. Given this present value, management should purchase the gold mine if the present value exceeds the purchase price P0 dollars. As illustrated by this example, real options theory applies derivatives pricing theory to any and all options embedded within a firm’s investment decisions. A classic textbook on this topic is Dixit and Pindyck (1994).

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Reduced-Form Models The second class of credit risk models are called reduced-form models. To understand the intuition behind the reduced-form model, it is best to start with Jarrow and Turnbull’s (1992, 1995) original formulation. The reduced-form model is an extension of the HJM model of Chapters 23–25. The reduced-form model uses all of the assumptions A1–A7 from the HJM libor model, with two modifications. One, assumption A2, no credit risk, is removed because the purpose of the reduced-form model is to price and hedge risky debt. The second modification is that the simple (default free) forward rates need not follow a lognormal distribution. Other evolutions are possible, and for the purposes of this section, we keep the evolution completely arbitrary. Note that unlike the structural model discussed in the previous section, reducedform models consider an economy with random interest rates. This is of particular importance in the risk management of fixed-income securities where interest rate and credit risk are paramount. We let rt denote the random default-free spot rate of interest (continuously compounded), and B(t, T) the time t value of a default-free zero-coupon bond paying a dollar at time T. In addition to assumptions A1–A7, the reduced-form model needs two more assumptions to characterize a firm’s risky debt. For easy comparison to the structural model, we consider a firm that has as one of its liabilities a zero-coupon bond with face value K and maturity date T. This assumption is easily relaxed in

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CHAPTER 26: RISK MANAGEMENT MODELS

a reduced-form model, unlike the complexity that arises when relaxing this same assumption in a structural model. The first additional assumption is as follows. A8. The firm’s zero-coupon bond trades continuously in time. Note that unlike the structural model, the reduced-form model only assumes that this particular debt issue trades. The firm’s other liabilities need not trade. This overcomes the key limitation of the structural model discussed previously. The second additional assumption characterizes the payoffs to the risky debt issue if the firm defaults. This characterization, therefore, must include two components (1) an assumption about when default occurs (i.e., specifying a random default time) and (2) an assumption about the loss rate on the debt issue in the case of default. A9. The firm’s default time 𝜏 is the first jump time of a Poisson process with intensity 𝜆 > 0. In the event of default, the loss rate is (1 − 𝛿), where 0 < 𝛿 < 1 is the debt’s recovery rate. The default intensity 𝜆 is the probability of default per unit time, given that the firm has not yet defaulted. The firm’s default time is denoted by 𝜏. The loss rate is given by the fraction (1 − 𝛿), where 𝛿 is the recovery rate. All of these parameters are easily estimated using historical data, a topic we leave to more advanced courses. The payoff to the debt can be written abstractly as K1{𝜏 >T} + 𝛿K1{𝜏 ≤ T}

(26.11)

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where 1(𝜏>T) =

𝜏>T 𝜏≤T

1 if {0 if

The symbol 1{𝜏 > T} is called an indicator function. It takes the value 1 if default does not occur by time T and the value 0 otherwise. Using this indicator function, expression (26.11) shows that the debt’s payoff is equal to K dollars if no default occurs prior to the debt’s maturity and 𝛿K dollars otherwise. The payoff in default is only 𝛿 < 1 percent of the original face value of the debt K. Let Lt denote the time t value of this debt issue. Under the no arbitrage assumption, just as in the HJM model, it can be shown that there exists risk-neutral probabilities 𝜋 such that all default-free zero-coupon bonds and the firm’s debt are martingales, after normalization by the value of a money market account. This implies that T

B (t, T) = E𝜋t (e− ∫t T

Lt = E𝜋t (k1{𝜏>T} e− ∫t

rs ds

r𝜎 ds

(26.2a)

) 𝜏

+ 𝛿K1{𝜏≤T} e− ∫t

rs ds

)

(26.12b)

THE FOUR RISKS

where the subscript “t” indicates the expectation is taken at time t. Expression (26.12a) is just the familiar risk-neutral valuation formula. The price of a defaultfree zero-coupon bond is just the expected discounted value of a dollar paid at time T, where one discounts using the spot rate of interest. Expectations are computed using the risk-neutral probabilities. Expression (26.12b) also uses risk-neutral valuation. This expression shows that the debt’s value is equal to its expected discounted payoff across the two possibilities regarding default: (1) the debt does not default before maturity, in which case it receives its face value K, and (2) default occurs prior to maturity, in which case the debt receives its recovery value. One discounts, as before, using the spot rate of interest, and expectations are computed using the risk-neutral probabilities. Using properties of the Poisson process and expression (26.12a), one can show that (see Jarrow 2009) T

̃ ̃ Lt = e−𝜆(T−t) KB (t, T) + ∫ 𝜆ẽ −𝜆(s−t) 𝛿KB (t, s) ds

(26.13)

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t

where 𝜆̃ is the risk-neutral default probability per unit time. We see here that the risky zero-coupon bond is worth less than the defaultfree zero-coupon bond KB (t,T). The first term represents the expected discounted ̃ value of the promised payment conditioned on no default. Here e−𝜆(T−t) is the riskneutral probability of no default. The second term represents the expected discounted ̃ value of the recovery value conditioned on default, where 𝜆ẽ −𝜆(s−t) is the risk-neutral probability of default at time s. One sums across all times s where default is possible, yielding the integral in the second term. The risk-neutral probability of default 𝜆̃ per unit time differs from the actual default probability 𝜆 by a risk premium. If default risk is idiosyncratic (firm specific), however, then the risk-neutral and actual default intensities are equal (see Jarrow 2009). After conditioning on the state of the economy, it is likely that what causes one firm to default versus another is idiosyncratic. It is an open research question as to whether this is true. Unlike the structural model, it is very easy to extend the reduced-form model to have more complex default processes, to price other traded liabilities is- sued by the firm, and to price credit derivatives. Extension 26.3 shows how to apply reducedform models to price credit default swaps that were introduced in Chapter 7.

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EXTENSION 26.3: Credit Default Swaps Credit default swaps are term insurance contracts written on traded bonds with a maturity date, a notional value, and a premium. If the insured bond defaults during the life of the contract, then the CDS seller pays off the difference between the notional and market value of the bond. Of course, the market value declines due to default. For this coverage, the CDS buyer pays a regular premium payment. The remainder of this extension shows how one can use the reduced-form model to price CDS. We use the same model as in the previous section and consider a CDS written on a bond issued by our firm with maturity date T*, coupon payments, and a notional value of $1. We denote the time t price of this coupon bond by Lt . We let 𝜏 be the firm’s random default time, and if the firm defaults prior to the coupon bond’s maturity date, we assume that the bond incurs a loss of (1 - 𝛿) dollars, where 𝛿 is the recovery rate. We can now value a CDS on this coupon bond. We let the CDS have a maturity date T < T* and a notional value equal to $1. In this CDS, the protection seller agrees to pay the protection buyer the loss on the debt if the firm defaults before the maturity date of the CDS. In return, the protection buyer pays a constant dollar spread c times the notional value at the fixed intermediate dates t, t + 1, … , T - 1. These payments continue until the swap’s maturity or the default date, whichever happens first. Using the risk-neutral valuation procedure, the time t value of the CDS to the protection seller is therefore T−1

CDSt = E𝜋t

(∑ k =t

c1{𝜏>k} e− ∫t

T

rs ds

T

− (1 − 𝛿) 1{𝜏≤T} e− ∫t

rs ds

)

(1)

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This is the present value of the premiums received (the summation term) less the losses paid if default occurs. It can be shown that this can be simplified to T−1

CDSt = c

∑

̃ −𝜆(k−t)

e

T

̃ B (t, k) − (1 − 𝛿) 𝜆̃ ∫ e−𝜆s (s − t) B (t, s) ds

k =t

(2)

t

where 𝜆̃ is the risk-neutral default probability per unit time. The market clearing CDS rate c is that spread that gives the CDS zero value, that is, ̃

c = (1 − 𝛿) 𝜆̃

∫t T e−𝜆s (s − t) B (t, s) ds T−1

≈ (1 − 𝛿) 𝜆̃

(3)

̃

∑

e−𝜆(k−t) B (t, k)

k =t

We see that the CDS rate is approximately equal to the risk-neutral expected loss (per year)—the risk-neutral default probability times the loss rate—on the firm’s debt. The CDS price takes into account risk aversion through the use of the risk-neutral probabilities. CDS trade on many firms’ debt issues, and the market uses CDS spreads and expression (3) to infer the market’s view of the risk of default as reflected in the implied risk-neutral default probability (for a given recovery rate). The reduced-form methodology is also easily employed to understand and price even more complex credit derivatives like collateralized debt obligations (see Jarrow 2009).

THE FOUR RISKS

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Liquidity Risk Liquidity risk corresponds to the relaxation of the competitive markets assumption. Recall that the competitive markets assumption implies that all traders act as price takers, that is, their trades have no impact on the buying or selling price. Liquidity risk occurs when there is a quantity impact of a trade on the price obtained. Quantity impact occurs in actual markets due to market power (size) or asymmetric information (which means the counterparties have differential information, e.g., a seller is likely to know more about a used car’s quality than the buyer). Market power is easy to understand. Buying a large quantity of any security changes the price because the buyer needs to induce the seller to sell. Asymmetric information causes a quantity impact because if one believes that the seller is informed and selling because the security is worth less than its current price, clearly the selling price must decline. In conjunction, these two reasons imply that selling (buying) large quantities of a security usually decreases (increases) the price obtained. When markets are volatile and disrupted, the quantity impact is larger than when markets are quiet and stable. The failure of Long-Term Capital Management (LTCM) in 1998 is perhaps the most infamous example of the importance of liquidity risk. LTCM was a hedged fund started by John Meriwether, ex–Salomon Brothers bond traders, and several finance academics in 1994. Using convergence arbitrage trading strategies in bond markets, they had phenomenal profit performance for the first four years of their life, providing returns in excess of 20 percent. A characteristic of their bond arbitrage trading strategies was that they required large leverage to transform small differences in bond spreads into large returns for their clients. This, of course, implies a small capital position. LTCM’s performance turned bad in May and June 1998, when mortgage-backed securities lost significant value. Then, Russia’s defaulting on its debt in August 1998, and the resulting market reactions, caused significant losses in LTCM’s asset value. This loss in asset value resulted in LTCM receiving margin calls for its futures positions and collateral calls for its swap positions. To obtain the additional capital, LTCM had to sell its assets. Selling assets when the market is spooked and declining is problematic. These asset sales resulted in increased value losses for LTCM as the process of selling large quantities of securities caused the market to drop further. A cascade of additional margin and collateral calls and market drops resulted, eventually leading to the failure of LTCM and the bailout of LTCM organized by the New York Federal Reserve Bank to avoid a financial market collapse (for a detailed history, see Jorion 2000). There are two approaches to modeling liquidity risk, which differ based on the permanence of the quantity impact of a trade on the price. If the quantity impact on the price is temporary, then liquidity risk is analogous to an endogenous transaction cost. By temporary, we mean that the quantity impact on the price only lasts for the trade under consideration, and after the trade, the price reverts back to its proper value. Çetin et al. (2004) were the first to study this approach in the context of derivatives pricing. Fortunately, the derivatives pricing methodology extends under this form of liquidity risk. Hence all of the results previously studied in Chapters 1–25 apply directly, with few, if any, changes.

677

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CHAPTER 26: RISK MANAGEMENT MODELS

If the quantity impact on the price is permanent, however, then liquidity risk takes on a different characterization. In this case, traders can act strategically and manipulate the price to their advantage. Here liquidity risk needs to include the consideration of market manipulation. Jarrow (1992, 1994) was to first to study manipulation in the context of derivatives pricing. Unfortunately, the standard derivatives pricing methodology fails to hold under this form of liquidity risk. Pricing and hedging become context specific, depending on the traders, their strategies, and the market setting. Here all of the results previously studied in Chapters 1–25 fail to apply, and new models are needed. The extensions in this case are the subject of current research.

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Operational Risk Operational (and legal) risk is the risk inherent in running a financial institution. These risks usually cause only small losses, but sometimes they have resulted in large blowouts. A stunning example of operational risk happened on December 8, 2005, when a broker at Japan’s Mizuho Securities Company transposed the quantity and the selling price in the company’s order processing system. He mistakenly typed an order to sell 610,000 shares of a small company J-Com at 1 yen instead of the other way round. The error was detected only a minute and a half later, but the Tokyo Stock Exchange’s all-electronic trading system thwarted all cancellation efforts. The result was a loss of $335 million.6 In February 7, 1991, traders at now-defunct Wall Street firm Salomon Brothers submitted an unauthorized customer bid in the amount of $1 billion as what Salomon termed a “practical joke” to tease an employee, but they failed to stop those bids on time, and an extra $1 billion worth of bonds got purchased. Eventually, the acquired securities were transferred to Salomon’s account at no loss to the firm (see Chatterjea 1993). Good management controls and procedures can minimize operational risk. Consequently, the control and minimization of this risk really falls under the heading of business management or organizational behavior and not finance. Nonetheless, for pricing and hedging this risk, finance is again useful. Operational risk can be modeled using the same mathematics as used in the reduced-form credit risk models. Indeed, operational risk can be viewed as a negative event, which if it occurs, generates a loss to the firm. This is exactly how the reducedform credit risk models view default. Consequently, the modeling techniques used in credit risk can be applied to operational risk (see Jarrow 2008; Jarrow et al. 2010a). The only substantive difference between these two models is that there is more publicly available data on credit risk events (realizations and losses) then there are on operational risk events. However, these differences are narrowing as various industry groups are creating operational risk databases.

6

See “Mizuho Settles Trade Debacle; Japanese Minister Scolds Brokers for Taking Advantage of Error,” Wall Street Journal, December 14, 2005.

THE FUTURE OF MODELS AND TRADED DERIVATIVES

26.6

The Future of Models and Traded Derivatives

The credit crisis of 2007 and its resulting aftermath created a discomfort with risk management models and derivatives. This discomfort raises the question as to the future of risk management models and derivatives. This section addresses this question, in light of what we have now studied in this book. The following discussion is based on Jarrow (2011).

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Model Risk When using financial risk management models, one must recognize that there is model risk. Model risk occurs because models are approximations—simplifications— of a complex reality. Statisticians Box and Draper (1987, p. 74) said, “Remember that all models are wrong: the practical question is how wrong do they have to be to not be useful.” A model’s usefulness—the quality of the approximation—needs to be judged relative to a purpose. Very crude models can be useful for decision making in some contexts, whereas more sophisticated models are needed in others. The usefulness of the model should be tested and continually validated to assure continued accuracy of the approximation. To understand how to test a model, we first need to divide a model’s assumptions into two types: robust and critical assumptions. A robust assumption is one in which the implications of the model only change slightly if the assumption is modified only slightly. In contrast, a critical assumption is one in which the implications of the model change discretely if the assumption is only changed slightly. This distinction is important because since models are approximations of a complex reality, we may not get the assumptions exactly correct. With robust assumptions, we do not need to worry too much. For small errors in the robust assumptions, the implications only change by a small amount. We need to be careful, however, with the critical assumptions. If we get the critical assumptions wrong, by just a little, the implications completely change. Now to test a model, one needs to test all the model’s relevant implications and all the assumptions, if possible. Sometimes, however, not all the assumptions can be tested. In this case, one must test all the critical assumptions. If any of the implications or critical assumptions are rejected, the model should not be used. Why? Because using an incorrect model generates poor risk management decisions. If a model is accepted for use, the implications and critical assumptions need to be continually monitored and validated because markets are dynamic and constantly changing. It is always possible that if the market’s structure shifts because of regulatory or institutional changes, the underlying assumptions may no longer be reasonable approximations. If this happens, then pricing and hedging will no longer work. The best way to understand models and their use is to consider an analogy. Models are analogous to medical prescription drugs. Prescription drugs have great medical benefits if used properly, with educated use. If used wrongly, however, prescription drugs can have negative consequences, even death. Because prescription drugs can

679

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CHAPTER 26: RISK MANAGEMENT MODELS

cause death if used incorrectly, this does not mean that we should stop using them. It does mean, however, that we need educated and regulated use. In fact, prescription drugs should probably be used more because they save and prolong lives. The same is true of models. Financial markets have become too complex to navigate without risk management models. Determining a price—fair value—is sometimes not an issue because in many cases, expert judgment may provide reasonable estimates. However, the following are true: ■ ■

There is no way to hedge a portfolio, that is, determine hedge ratios, without a model.

■

There is no way to price a derivative in an illiquid market without a model.

These issues are at the heart of risk management. Hence financial risk management models are here to stay.

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Derivatives Derivatives have existed for thousands of years. They have continued to exist (trade) across different centuries, across different governments, and across different economic systems. Despite occasional uproars about their evil uses, derivatives have continued to trade because they help complete markets and, consequently, improve societal welfare. If history shows us nothing else, it shows us that derivatives are here to stay! Given this insight, the best way to live with derivatives is through education and knowledge. We have started you on this journey to understanding with the content of this book, but this is just the beginning! There is much more to learn and master. We have just walked with you on only the first part of your journey to understanding derivatives, and it is now time for us to leave. However, we leave you knowing that you are well prepared for the obstacles ahead.

26.7

Summary

1. A firm’s risk management decisions can be understood using the firm’s balance sheet and computing the firm’s loss distribution defined as Prob{ΔAt − ΔLt ≤ x}, where the time t asset’s value is At , the time t liability is Lt , and ΔAt ≡ At+Δt − At and ΔLt ≡ Lt+Δt − Lt correspond to the change in the value of the assets and liabilities, respectively, over the interval [t, t + Δt]. 2. Value-at-risk (VaR𝛼 ) for an 𝛼 loss probability is that number such that Prob{ΔAt − ΔLt ≤ −VaR𝛼 } = 𝛼. In some cases, VaR𝛼 penalizes diversification, which is a failing in this risk management statistic. 3. Scenario analysis (which also goes by the popular name stress testing) starts with a selection of a particular set of scenarios (or states of the economy) that the firm wants to protect against. For these scenarios, the firm wants to compute the losses realized on its assets and liabilities to see if it has sufficient equity capital to avoid

QUESTIONS AND PROBLEMS

insolvency. Scenario analysis’s major failing is that it does not assign probabilities to the various scenarios. 4. Credit risk is the risk that a counterparty will fail to execute on the terms of a contract. There are two models for quantifying credit risk: structural and reducedform models. The structural model is useful for internal management decision making and understanding the economics of the firm’s balance sheet, whereas the reduced-form model is useful for pricing and hedging credit-risky instruments trading in the financial markets. 5. Both liquidity and operational risk can be modeled using the techniques studied in this book. Operational risk uses the same models as credit risk. 6. Models are needed for risk management in complex financial markets. To avoid model misuse, better education on models is needed.

26.8

Cases

Amaranth Advisors: Burning Six Billion in Thirty Days (Richard Ivey School of

Business Foundation Case 908N03-PDF-ENG, Harvard Business Publishing). This case studies the staggering losses encountered by a hedge fund and introduces concepts like liquidity risk, value-at-risk, spread trades, and the use of derivatives.

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Delphi Corp. and the Credit Derivatives Market (A) (Harvard Business School

Case 210002-PDF-ENG). This case considers the use of credit derivatives to hedge credit risk and/or to speculate on the price of corporate debt and also how credit derivatives affect the incentives of creditors to negotiate with a distressed company attempting to avoid bankruptcy. CITIC Tower II: The Real Option (University of Hong Kong Case HKU199-PDF-

ENG, Harvard Business Publishing). This case studies real options involved in the decision to purchase newly reclaimed land in Hong Kong for the purpose of building an office building at an appropriate time in the future.

26.9

Questions and Problems

26.1. When using value at risk at the 5 percent level to determine equity capital,

do the debt holders need to worry about the magnitude of the losses at the 1 percent level? Explain. 26.2. Consider a firm whose business is to purchase distressed debt, that is, bonds

of firms that have defaulted, hoping to profit from the bankruptcy resolution process. Which of the four risks (market, credit, liquidity, and operational) does this distressed debt investment firm consider when evaluating the distressed debt? Explain.

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682

CHAPTER 26: RISK MANAGEMENT MODELS

26.3. Suppose you enter into a $5 bet with a friend on the winner of a soccer game.

What risks do you face in this gamble? 26.4. In the structural credit risk model, it is claimed that the debt holders own the

firm, and not the equity holders. Explain why. 26.5. Since the equity holder’s position is equivalent to a call value on the firm’s

assets, and the equity holders manage the firm’s assets, can they increase the value of their equity by increasing the firm’s asset volatility? Explain your answer. 26.6. What are the three risks faced by a bond in the reduced-form credit risk

model? 26.7. Which risks caused the hedge fund Long-Term Capital Management to fail? 26.8. Explain why operational risk and credit risk can be modeled using similar

techniques. 26.9. What assumption makes the structural model not useful for actual pricing and

hedging but makes the reduced-form model appropriate for these activities? 26.10. Your grandmother gives you an old silver flower vase as a birthday gift. You

want to sell it to buy a new computer. What risks do you face with respect to this asset? 26.11. Internet markets—like eBay—decrease which risk of holding assets? 26.12. Why are models analogous to prescription medical drugs? Are derivatives

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likely to discontinue trading? Explain. The next two questions are based on the following data for three companies.

Expected Return, Volatility, and Portfolio Weights Ali Co. (Company 1)

Barb Co. (Company 2)

Cao Co. (Company 3)

Weekly expected return

0.0020

0.0025

0.0030

Weekly volatility of the returns

0.03

0.04

0.05

Portfolio weights

1/3

1/3

1/3

QUESTIONS AND PROBLEMS

Correlation of Expected Returns Ali Co. (Company 1)

Barb Co. (Company 2)

Cao Co. (Company 3)

1

0.4

0.8

Barb Co.

0.4

1

0.5

Cao Co.

0.8

0.5

1

Ali Co.

26.13. Compute the weekly 95 percent and 99 percent value-at-risk for Ali Co. 26.14. Compute the weekly 95 percent and 99 percent value-at-risk for a portfolio

consisting of the three stocks in equal proportions. 26.15. Now that you have completed the book and are on your way to mastering

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derivatives, briefly discuss your views as to whether derivatives are good or bad? Feel free to use the internet to get some quotes to support your answer.

683

Appendix A Mathematics and Statistics A.1 Exponents and Logarithms Exponents

A.3 The Normal Distribution A.4 A Taylor Series Expansion

Logarithms

Single Variables

Continuous Compounding

Multiple Variables

A.2 The Binomial Theorem

EXPONENTS AND LOGARITHMS

A.1

Exponents and Logarithms

Exponents and logarithms are useful for valuing cash flows across time and for computing continuously compounded returns.

Exponents Arithmetic classes taught us that 2 times 2 is 4, and algebra classes showed us that we can write this as 22 (read “2 raised to the power of 2”). Here 2 is an exponent denoting the number of times 2 multiplies itself. We can generalize this example and write “x multiplied by x” as x2 . We can generalize further to denote “n products of x” as xn ≡ x × x … n times

(1.1)

The ≡ sign denotes that “this is true by definition.” An exponent may also be a fraction that denotes a root or a radical. For example, x1/2 means x1/2 × x1/2 = x; we also write x1/2 as √x and read it as “square root of x ”. n When generalized, x1/n is the “nth root of x,” and it is also expressed as √ x, which means x1/n × x1/n … n times ≡ x (1.2) Negative exponents are used to create an inverse. For example, 2−1 is 1/21 = 1/2. This is generalized to define x−n ≡ 1/(x × x … n times) ≡ 1/xn

(1.3)

Of course, the preceding expression is only defined when x is not equal to zero (in notations, x ≠ 0); moreover, x0 = 1, when x ≠ 0. These puzzling results can be explained by the law of exponents. You can write x raised to the power 0 as x0 = xm−m = xm × x−m = xm /xm = 1 This holds true when x does not equal zero. When x = 0 and we raise it to the power 0, then we have zero in the denominator, and we are essentially dividing by 0, which is forbidden by mathematicians because it gives nonsensical results. Next we write some useful properties of exponents (see Table A.1). We state the laws and give an example illustrating their use.

Logarithms We only consider natural logarithms in this book (see Table A.2). The natural logarithm, written as log(x), is defined by log (x) = y is equivalent to ey = x

(1.4)

685

686

APPENDIX A: MATHEMATICS AND STATISTICS

TABLE A.1: Laws of Exponents Laws of Exponents

Examples

1. xm xn = xm+n 1 2. x−m = m x xm 3. xm−n = n x

5. (xm )n = xmn

x2 x3 = x5 1 x−2 = 2 x x5 x5−3 = 3 = x2 x x5 0 5−5 x =x = 5 =1 x (x2 )3 = (x2 ) (x2 ) (x2 ) = x6

6. xm ym = (xy)m and xm /ym = (x/y)m

x5 y5 = (xy)5 and x4 /y4 = (x/y)4

4. x0 = xm−m = xm

1 = 1, when x ≠ 0 xm

TABLE A.2: Laws of Logarithms Laws of Exponents

Examples

1. log(xy) = log x + log y

log 50 = log 5 + log 10

2. log(x/y) = log x – log y

log(10/2) = log 10 − log 2 = log 5 + log 2 − log 2 = log 5

3. log(x)n = n log x

log 23 = log 8 = log(2 × 2 × 2) = 3 log 2 and log 21/5 = (1/5) log 2

4. log 1 = 0

log 1 = 0 because e0 = 1

where e is the Euler number, approximately 2.711828, defined as the limit 1 n e = lim (1 + ) n→∞ n

(1.5)

when n goes to infinity or as the infinite series ex = 1 +

x x2 x3 xn + + +⋯+ +⋯ 1 1×2 1×2×3 n!

(1.6)

for any real number x. Notice that a representative term in expression (1.6) is given by the (n + 1)th term xn /n!, where n! is read as n-factorial and is defined by n! = n × (n − 1) × … × 3 × 2 × 1 and 0! = 1.

THE BINOMIAL THEOREM

Continuous Compounding If X dollars is invested today at an interest rate r that is compounded m times per year, and m becomes infinitely large, then it has the time T value X[(1 + r/m)m ]T = X[(1 + r/m)m/r ]rT → XerT

(1.7)

because (1 + r/m)(m/r) ≡ [1 + 1/(m/r)](m/r) = (1 + 1/n)n goes to e as n = m/r goes to infinity. This can be used to model dividend yields. Consider buying one unit of a stock that pays dividends at a continuous rate of 𝛿 per year. Assume that the dividends are reinvested back into the asset (see section 12.3). To model this, we start by dividing the year into m periods of equal length and assume that a dividend of 𝛿/m times the stock value is paid at the end of each interval. At each payment date, we use the dividend to buy more of the stock. Consequently, the value of the position in the stock grows just like compound interest. Here, 𝛿/m is compounded m times per year, where m becomes infinitely large. Then, our one unit in the stock grows to [(1 + 𝛿/m)m ]T ≡ [(1 + 𝛿/m)(m/𝛿) ]𝛿T = e𝛿T

units after T years

(1.8)

This happens because (1 + 𝛿/m)(m/𝛿) ≡ [1 + 1/(m/𝛿)](m/𝛿) = (1 + 1/n)n goes to e as n = m/𝛿 goes to infinity.

A.2

The Binomial Theorem

One of the first facts you learn in algebra is how to square the sum of two variables x and y: (x + y)2 = x2 + y2 + 2xy The expression (x + y) is called a binomial because it is the sum of two terms. Next comes the cubic expansion of a binomial: (x + y)3 = x3 + 3x2 y + 3xy2 + y3 One can expand this binomial for an arbitrary integer n. The binomial theorem provides a method for doing this: (x + y)n =

n n 0 n n−1 1 n n−j j n xy + x y ⋯+ x y ⋯+ x1 y(n−1) (0) (1) (j) (n − 1) n

n 0 n n n−j j + xy = x y ∑ (j) (n) j=0

(2.1)

687

688

APPENDIX A: MATHEMATICS AND STATISTICS

where the binomial coefficient (which is the number of ways of selecting j things out of a possible number of n things) can be expressed as n n! = ( j ) j! (n − j) !

(2.2)

Here the expression n! is called n-factorial and is defined as n! = n × (n − 1) × (n − 2) × … × 2 × 1

(2.3)

For example, 4! = 4 × 3 × 2 × 1 = 24 and 1! = 1, and by definition, 0! = 1. Moreover, n! = n(n − 1)! = n(n − 1)(n − 2)!, and so on.

A.3

The Normal Distribution

A standard normal distribution function has a mean equal to 0 and a standard deviation equal to 1. The normal distribution function is denoted by N(d). This gives the probability that the underlying normal random variable z will take a value less than or equal to d, that is, N(d) = Prob(z ≤ d). This probability can be obtained from a normal distribution table (see Table A.3) or using computer software (e. g., spreadsheets). Usually, normal distribution tables give probabilities for z greater than 0. The normal density represents the mathematical derivative of the normal distribution function. The normal density function is symmetric around mean 0 (see Figure A.1) and is shaped like a bell. Because the normal density function is symmetric around 0, if you want to compute N(−d) for some negative value, use N(d) + N(−d) = 1.

EXAMPLE A.1: Cumulative Standard Normal Distribution ■

From the cumulative standard normal distribution table (Table A.3), N (0.75) = 0.7734 This is the probability that a normal random variable will have a value of 0.75 or less.

■

Likewise, N (0.55) = 0.7088

■

Or you can use spreadsheet programs like Excel or Lotus Notes to compute cumulative normal. Excel will use the command = NORMSDIST(z), where z is the underlying normal random variable (in Lotus Notes, use the command @NORMAL(z,0,1,0)).

THE NORMAL DISTRIBUTION

FIGURE A.1: Normal Density Function Probability density

N(d)

–∞

0

∞

d

Value

TABLE A.3: Cumulative Normal Distribution Function 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0

0.5

0.504

0.508

0.512

0.516

0.5199

0.5239

0.5279

0.5319

0.5359

0.1

0.5398

0.5438

0.5478

0.5517

0.5557

0.5596

0.5636

0.5675

0.5714

0.5753

0.2

0.5793

0.5832

0.5871

0.591

0.5948

0.5987

0.6026

0.6064

0.6103

0.6141

0.3

0.6179

0.6217

0.6255

0.6293

0.6331

0.6368

0.6406

0.6443

0.648

0.6517

0.4

0.6554

0.6591

0.6628

0.6664

0.67

0.6736

0.6772

0.6808

0.6844

0.6879

0.5

0.6915

0.695

0.6985

0.7019

0.7054

0.7088

0.7123

0.7157

0.719

0.7224

0.6

0.7257

0.7291

0.7324

0.7357

0.7389

0.7422

0.7454

0.7486

0.7517

0.7549

0.7

0.758

0.7611

0.7642

0.7673

0.7704

0.7734

0.7764

0.7794

0.7823

0.7852

0.8

0.7881

0.791

0.7939

0.7967

0.7995

0.8023

0.8051

0.8078

0.8106

0.8133

0.9

0.8159

0.8186

0.8212

0.8238

0.8264

0.8289

0.8315

0.834

0.8365

0.8389

1

0.8413

0.8438

0.8461

0.8485

0.8508

0.8531

0.8554

0.8577

0.8599

0.8621

1.1

0.8643

0.8665

0.8686

0.8708

0.8729

0.8749

0.877

0.879

0.881

0.883

1.2

0.8849

0.8869

0.8888

0.8907

0.8925

0.8944

0.8962

0.898

0.8997

0.9015

1.3

0.9032

0.9049

0.9066

0.9082

0.9099

0.9115

0.9131

0.9147

0.9162

0.9177

1.4

0.9192

0.9207

0.9222

0.9236

0.9251

0.9265

0.9279

0.9292

0.9306

0.9319

1.5

0.9332

0.9345

0.9357

0.937

0.9382

0.9394

0.9406

0.9418

0.9429

0.9441

(Continued )

689

690

APPENDIX A: MATHEMATICS AND STATISTICS

TABLE A.3: (Continued ) 1.6

0.9452

0.9463

0.9474

0.9484

0.9495

0.9505

0.9515

0.9525

0.9535

0.9545

1.7

0.9554

0.9564

0.9573

0.9582

0.9591

0.9599

0.9608

0.9616

0.9625

0.9633

1.8

0.9641

0.9649

0.9656

0.9664

0.9671

0.9678

0.9686

0.9693

0.9699

0.9706

1.9

0.9713

0.9719

0.9726

0.9732

0.9738

0.9744

0.975

0.9756

0.9761

0.9767

2

0.9772

0.9778

0.9783

0.9788

0.9793

0.9798

0.9803

0.9808

0.9812

0.9817

2.1

0.9821

0.9826

0.983

0.9834

0.9838

0.9842

0.9846

0.985

0.9854

0.9857

2.2

0.9861

0.9864

0.9868

0.9871

0.9875

0.9878

0.9881

0.9884

0.9887

0.989

2.3

0.9893

0.9896

0.9898

0.9901

0.9904

0.9906

0.9909

0.9911

0.9913

0.9916

2.4

0.9918

0.992

0.9922

0.9925

0.9927

0.9929

0.9931

0.9932

0.9934

0.9936

2.5

0.9938

0.994

0.9941

0.9943

0.9945

0.9946

0.9948

0.9949

0.9951

0.9952

2.6

0.9953

0.9955

0.9956

0.9957

0.9959

0.996

0.9961

0.9962

0.9963

0.9964

2.7

0.9965

0.9966

0.9967

0.9968

0.9969

0.997

0.9971

0.9972

0.9973

0.9974

2.8

0.9974

0.9975

0.9976

0.9977

0.9977

0.9978

0.9979

0.9979

0.998

0.9981

2.9

0.9981

0.9982

0.9982

0.9983

0.9984

0.9984

0.9985

0.9985

0.9986

0.9986

3

0.9987

0.9987

0.9987

0.9988

0.9988

0.9989

0.9989

0.9989

0.999

0.999

3.1

0.999

0.9991

0.9991

0.9991

0.9992

0.9992

0.9992

0.9992

0.9993

0.9993

3.2

0.9993

0.9993

0.9994

0.9994

0.9994

0.9994

0.9994

0.9995

0.9995

0.9995

3.3

0.9995

0.9995

0.9995

0.9996

0.9996

0.9996

0.9996

0.9996

0.9996

0.9997

3.4

0.9997

0.9997

0.9997

0.9997

0.9997

0.9997

0.9997

0.9997

0.9997

0.9998

3.5

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

0.9998

3.6

0.9998

0.9998

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

3.7

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

3.8

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

0.9999

3.9

1

1

1

1

1

1

1

1

1

1

4

1

1

1

1

1

1

1

1

1

1

A TAYLOR SERIES EXPANSION

A.4

A Taylor Series Expansion

Formally introduced by the English mathematician Brook Taylor in 1715, a Taylor series allows us to express special functions as an infinite sum of terms involving polynomials.

Single Variables The Taylor series expansion of a single variable function f (x) that is infinitely differentiable in a neighborhood of a number a is given by the power series f (x) = f (a) +

f 1 (a) f (n) (a) f 2 (a) (x − a) + (x − a)2 + ⋯ (x − a)n + ⋯ 1! 2! n!

(4.1)

where n! = n × (n − 1) × … × 2 × 1 denotes the factorial of n and f (n) (a) denotes the nth derivative of f evaluated at the point a. Notice that f 0 is defined to be f itself and (x − a)0 and 0! are both defined to be 1; f 1 and f 2 are customarily written as f ′ and f ′′, respectively. Consider the function graphed in Figure A.2. A Taylor series approximation of this function at the point x = a using just the first two terms is given by f (x) ≈ f (a) + f ′ (a) (x − a)

(4.2)

Algebra tells us that the right side of this expression is a straight line tangent to the function f (x) at x = a. Expression (4.2) shows that this straight line (shown by a dashed line in the figure) is an approximation to the function f (x) at x = a.

FIGURE A.2: Taylor Series Approximation y y = f(x)

f´(a)

f(a)

0

y = f(a) + f´(a)(x – a)

a

x

691

692

APPENDIX A: MATHEMATICS AND STATISTICS

Such approximations work well for small changes in x around x = a. A better approximation for larger changes in the variable x is obtained by retaining the second derivative as given by f (x) ≈ f (a) + f ′ (a) (x − a) +

1 f ′′ (a) (x − a)2 2

(4.3)

Multiple Variables The Taylor series for a well-behaved function of multiple variables can be expressed as ∞

∞

𝜕n1 +⋯+nd f (x1 − a1 )n1 ⋯ (xd − ad )nd (a1 , … , ad ) n n ∑ ∑ n1 ! … nd ! ( 𝜕x11 ⋯ 𝜕xdd ) n1 =0 nd =0 (4.4) To illustrate this general result, consider a function f of two variables x and y. The Taylor series expansion to the second order about the point (a, b), writing (x − a) as Δx, and (y − b) as Δy, is f (x1 , … , xd ) =

⋯

f (x, y) ≈ f (a, b) +

𝜕f (a, b) 𝜕f (a, b) Δx + Δy 𝜕x 𝜕y

2 2 𝜕2 f (a, b) 1 𝜕 f (a, b) 2 𝜕 f (a, b) 2 + Δx + Δy + 2ΔxΔy 𝜕x𝜕y ] 𝜕y2 2! [ 𝜕x2

(4.5a)

Writing f (x, y) − f (a, b) as Δf, and moving the first term on the right side of this expression to the left side gives the simpler expression Δf ≈

2 2 𝜕2 f 𝜕f 𝜕f 1 𝜕f 2 𝜕f 2 Δx + Δy + Δx + Δy + 2ΔxΔy 𝜕x 𝜕y 𝜕x𝜕y 𝜕y2 2! 𝜕x2

These results are used in Chapters 20 and 25.

(4.5b)

Glossary

Algorithmic trading or algo Trading using an algorithm (which is a set of rules or sequence of steps) to identify patterns in real-time market data and exploit potentially profitable trading opportunities.

Basel Committee on Banking Supervision Established in 1974 by the Group of Ten countries’ central-bank governors; formulates broad supervisory standards and recommends statements of best banking practice.

American depository receipt A US exchange-traded security that has a fixed number of foreign stock shares as the underlying.

Basis The spot price minus the futures price.

American option An option that can be exercised anytime during its life. Annual Percentage Rate (APR) The restatement of an interest rate (e.g., continuously compounded) in terms of an annualized simple interest rate. Arbitrage A trading strategy that has a chance to make riskless profits without requiring a net investment of funds. Arbitrageur Trader who attempts to make arbitrage profits from price discrepancies. Arbitration A system of mediating disputes before going to the courts. Ask (or offer) price The price at which a dealer offers to sell securities to a customer. Asset-backed security (ABS) Bond issued by a specialized purpose vehicle (SPV) that is backed by assets as collateral. At-the-money (option) An option whose underlying asset’s price is close to the exercise price. Audit trail A detailed information record of a financial security transaction. Backward induction A mathematical technique for solving a problem where one starts at the last date in a model and obtains the solution by working backward through time. Backwardation A forward or futures market where the forward or futures price is lower than the spot price, respectively. Banging (or marking) the close Manipulating prices by trading a large security position leading up to the market close followed by offsetting the position before the end of trading. Banker’s discount yield The rate used for quoting Treasury bills.

Basis point 1 percent.

(or

bp) One-hundredth (1/100th) of

Bearish trade A trade benefiting if security prices decrease in value. Bid price The price at which a dealer offers to buy securities from a customer. Binomial stock price evolution A model of a stock price evolution using a tree in which from any point in time onward, the stock price takes only one of two possible values. Block trade Stock trade involving ten thousand shares or more. Bond or fixed-income security The debt issued by a borrower who promises both to pay interest on the borrowing and to repay the principal borrowed at some future date (see consols for an exception). Bond-equivalent yield The holding period return earned on a bond if held until maturity. Bootstrap method The technique of creating a zerocoupon bond price term structure from market data (including computation of any missing prices by interpolation). Broker A financial intermediary who matches buyers and sellers and earns commissions for this service. Bullish trade A trade benefiting if security prices increase in value. Butterfly spread An option trading strategy created by trading four options of the same type (all calls or all puts) with three different strike prices: two options with extreme strike prices are purchased (written) and two options are written (purchased) with middle strike prices. Buying hedge (or long hedge) Taking a long position in a forward or a futures contract to hedge the price risk from a spot market purchase. Buy–write (option) Simultaneously buying the underlying asset and selling the call. 693

694

GLOSSARY

Calibration A mathematical technique that estimates a derivative pricing model’s parameters by equating the model’s price to the market price. Call option A financial security that gives the owner the right to buy a specified quantity of a financial or real asset on or before a fixed future date by paying an exercise price agreed on when the contract is written. Callable bond A bond whose issuer has the option to redeem the bond before maturity. Cap A portfolio of caplets. Capital gain or loss (Gain) Realized by selling an asset at a price higher than the purchase price. (Loss) Realized by selling an asset at a lower price than the buying price. Caplet A European call option on an interest rate. Cash settlement Ending a derivatives contract by making a cash payment instead of exchanging physical securities. Cash-and-carry argument An argument used to determine the forward price where one purchases the underlying commodity with borrowed cash and holds the commodity until the forward contract’s maturity date. Central limit theorem (CLT) A key theorem in probability that states that the distribution for the average of a collection of independent random variables, when “standardized” (by subtracting from the average the common mean and dividing the expression by the standard deviation of the average), approaches a standard normal distribution as the sample size increases. Cheapest-to-deliver bond An embedded option available to the seller of a Treasury bond futures contract by which one can deliver any one of several Treasury bonds at expiration. Clearing (a trade) When buy and sell orders are matched, the trade is officially recognized, and the trade is recorded by the exchange’s clearinghouse. Clearing member Exchange member (broker and dealer) who clears trades. Clearinghouse An entity associated with or part of an exchange that clears trades. Close (futures market) The end of a day’s trading session. Closed-end fund Investment company that takes cash for the issuance of shares at inception, but no additional shares are issued thereafter. Closing transaction (or a reversing trade) A trade that cancels an outstanding open position. Collar (Option) A portfolio that combines a long position in the underlying asset with a long put and short call (with

different strike prices) so that the resultant portfolio has truncated payoffs on both the up and down sides. (Market regulation) A trading restriction that prohibits the buying or selling of stocks when the stock price moves by more than a certain number of points. Collateralized debt obligation (CDO) An asset-backed security whose collateral pool consists of ABS bonds, as distinct from mortgage loans. CDO bonds are partitioned into seniority tranches that specify the interest and principal payments the bonds receive. Collateralized debt obligation squared (CDOˆ2) An asset-backed security whose collateral pool consists of CDO bonds. Combination strategy A portfolio of options of different types on the same underlying asset with the same expiration date where the options are either all purchased or written. Commercial paper A promissory note, issued by a company, that matures in 270 days or less and that is sold as a zero-coupon bond. Commercial trader (futures) Futures trader such as a farmer, manufacturer, or commodities dealer who has legitimate hedging needs because of her commercial activity with the underlying commodity. Commitment of Traders (COT) A weekly report published by the Commodity Futures Trading Commission that shows aggregate commercial and noncommercial trader positions in certain futures and options markets. Commodity (US futures contract) The CFTC Act of 1974 defines a commodity to include any “goods and articles . . . and all services, rights, and interests” on which contracts for future delivery can be written. Commodity Futures Trading Commission (CFTC) The US federal government agency regulating futures and futures on option markets. Commodity indexed note A derivative whose holder receives a payment tied to the return on a physical commodity or a commodity index. Commodity pool (or commodity fund) Investment fund investing in commodity futures. Commodity swap A swap in which one counterparty makes a floating payment that is tied to the changes in a commodity price or the return on a commodity price index. Competitive market An idealized description of a market populated by a large number of traders who are too insignificant to influence prices.

GLOSSARY

Complete market A market where sufficient securities trade so that investors can obtain any desired future payoffs. Condor spread An option portfolio created by trading four options of the same type (all calls or all puts) with four different strike prices: two options with extreme strike prices are purchased (written) and two options are written (purchased) with middle strike prices. Consol A bond that never pays back the principal but pays interest at a promised rate forever. Consumer Price Index (CPI) An index measuring the cost of a typical household’s goods and purchased services. Contango A forward or a futures market where the forward or futures price is higher than the spot price. Convenience yield A benefit that accrues to the holder of a physical commodity. Convertible bond A bond whose holder has the option to convert the bond into a fixed number of shares of the borrowing company’s stock. Cost-of-carry model A model that determines a forward price of a commodity under the assumption that buying and carrying the commodity to the future has interest as the only cost. Counterparties The traders (buyer and seller) in a financial transaction. Coupon A periodic interest payment made on a bond. Coupon bond A bond that pays interest (coupons) periodically to the bondholder. Covered (option trade) An option trade that is accompanied by a position in the underlying asset that reduces the overall risk; the special case of a cash-secured put (a short put trade in which the writer deposits an amount equal to the strike price with the broker) is also considered a covered trade. Credit default swap A derivative security that provides insurance against the event that the issuer of a bond might default and the bond declines in value. Credit swap A swap in which both payments are based on individual stock returns in the same industry but with different credit ratings. Cross-hedge (imperfect hedge) A hedge in which the spot and futures positions do not exactly offset each other. Cross-trade An offsetting or noncompetitive match of a buy and sell order; normally banned, but permitted in special cases.

695

Cum-dividend The price of a stock before it goes ex-dividend. Currency swap A swap involving an exchange of payments in one currency for that in another. A plain vanilla currency swap (or a cross-currency swap) between two counterparties involves (1) an exchange of equivalent amounts in two different currencies on the start date, (2) an exchange of interest payments on two currency loans at intermediate dates, and (3) a repayment of the principal amounts on the loans at the ending date. Current assets These are cash or cash-equivalent assets held for collection, sale, or consumption within the enterprise’s normal twelve-month operating cycle; all other assets are noncurrent. Current liabilities The liabilities settled within the enterprise’s normal twelve-month operating cycle, or those held for trading, or those for which the entity does not have an unconditional right to defer payment beyond twelve months; other liabilities are noncurrent. Daily settlement (futures) Crediting/debiting of the day’s gains/losses to a futures contract’s margin account. Dark pool (or dark pool of liquidity) Secretive trade matching networks for institutional traders that do not send orders directly to an exchange or that are not displayed in a limit order book. Day order A limit order that gets cancelled if it is not executed by the end of the trading day during which it is placed. Day trader Trader who tries to profit from daily price movements, opening trading positions in the morning and closing them out at night. Dealer A financial intermediary who posts prices at which she buys (wholesale or bid price) or sells (retail or ask price) securities. Default A failure to pay a promised payment on a financial contract at the promised time. Delivery date (or maturity date) A predetermined date on which a seller of a futures contract agrees to deliver the underlying commodity to the buyer in exchange for the futures price. Delivery period (futures) Time period during which delivery of the underlying commodity, as stipulated in a futures contract, can occur. Delta The sensitivity of an option price to changes in the underlying asset price.

696

GLOSSARY

Delta-neutral A portfolio whose delta is zero. Derivative (or a derivative security) A financial contract that derives its value from an underlying variable such as a stock price, a commodity price, or even an interest rate. Diagonal spread A portfolio obtained by buying an option with a particular expiration date and strike price and simultaneously selling an option with a different expiration date and different strike price, where the options (both calls or both puts) have the same underlying asset.

Euribor (Euro Interbank Offered Rate) An interest rate index computed on Euro Interbank term deposits. Eurobond An international bond that is denominated in a currency not native to the issuing country. Eurocurrency Time deposits in a bank denominated in a currency outside the borders of the country where the bank is located. Eurodollar Dollar-denominated time deposits held in a bank outside the United States.

Discriminatory auction An auction where the successful bidders pay their bid prices to purchase the item being sold.

European option An option that can be exercised only at the expiration date.

Diversifiable risk A risk that can be eliminated in a portfolio of securities via diversification.

Exchange A physical or electronic location where buyers and sellers meet to trade a standardized commodity under a given set of rules.

Dividend Payment in the form of cash or stocks made to existing stockholders. Dividend yield A stock’s dividend payment expressed as a fraction of the stock price. Dollar return The value of an invested dollar on a future date. Dual trading Allowing a market participant to act as a broker and dealer on the same day, though not on the same transaction. Duration The weighted average of time to payment of a bond’s cash f lows where the weights correspond to the present value of the cash f lows received on the respective dates. Dynamic hedging A hedge that is regularly adjusted over time to maintain a riskless position. Efficient market A market in which security prices “fully and accurately” reflect all relevant information. Electronic communications network (ECN) An electronic system that automatically matches publicly displayed buy and sell orders at specified limit order prices. Employee stock options Call options on the stock that are granted by a company to employees as a form of noncash compensation. Equilibrium model Model that determines prices by requiring the equality of supply and demand. Equity option Option that has common stock as the underlying. Equity swap A swap in which one side makes a floating payment tied to the return on a stock or a stock index. Equity-linked note (ELN) A combination of a zerocoupon (or a low-coupon) bond with a return based on the performance of a single stock, a basket of stocks, or a stock index.

Exchange of physicals (EFP) Terminating a futures contract with a spot trade in which the buyer and the seller privately negotiate the delivery of the commodity at a location different from those in the futures contract. Exchange rate The price of one currency in terms of another. Exchange-traded fund (ETF) Investment company that gives shareholders fractional ownership rights over a basket of securities. Exchange-traded note (ETN) Bond issued by an underwriting bank, which promises to pay a return (minus any fees) based on the performance of a market benchmark (like a commodity price index) or some investment strategy. Ex-dividend date The date after which the stock owner does not get the last announced dividend. Execution A trade is executed when the counterparties agree on the terms, conditions, price, and quantity and commit to transact. Exercise price (or strike price) The predetermined price that an option holder pays to exercise the option. Exotic (derivative) Complex derivatives that are not ordinary call or put options. Expectations hypothesis The hypothesis that zerocoupon bonds of different maturities are perfect substitutes. Expense ratio (mutual fund) A mutual fund’s annual operating expenses expressed as a percentage of average net assets. Expiration date (option) The last day that an option holder gets to exercise the option. Extendible bond A bond whose holder has the option to extend the bond’s life.

GLOSSARY

Fail to deliver In a repurchase agreement, when a counterparty fails to deliver the securities, the loan is extended by a day at no extra interest. Federal funds market A borrowing or lending market for cash among US banks. Federal Reserve System (the Federal Reserve or simply “the Fed”) The central bank of the United States. Financial Accounting Standards Board (FASB) A private, not-for-profit organization whose primary purpose is to establish financial accounting standards that govern the preparation of financial reports by nongovernmental entities. Financial assets Paper assets such as stocks, bonds, and currencies that represent claims to real assets. Financial engineering The science of applying engineering tools to develop financial contracts to meet the needs of an enterprise. Financial futures (or financials) Futures contracts based on financial assets (like foreign currencies) or financial variables (like interest rates). Financial Industry Regulatory Authority (FINRA) A self-regulatory organization created by combining the self-regulatory functions of the New York Stock Exchange and the National Association of Securities Dealers. Fixed price offering An offer to sell securities where the price, the interest rate, and the maturity are announced beforehand. Floor A portfolio of floorlets. Floor broker A broker working on an exchange floor. Floorlet A European put option on an interest rate. Forex swap (or FX swap) A swap in which (1) the contract starts with an exchange of one currency for another and (2) it ends with a reverse exchange of these cash payments, with interest, at a subsequent date. Forward contract A binding agreement between a buyer and a seller to trade some commodity at a fixed price at a later date. Forward price (or delivery price) The fixed price in a forward contract in which the counterparties agree to trade a commodity on a future delivery date. Forward rate The rate one can contract today on riskless borrowing or lending over a future time period. Forward rate agreement A derivatives contract that pays off based on the interest rate for borrowing (or lending) over some future time period.

697

Forward rate curve The graph of forward rates as a function of the time to maturity. Front running Trading based on nonpublic information about an impending transaction by another person. Fundamental analysis Security analysis that involves reading accounting and financial information about a company to determine whether the share is overvalued or undervalued. Futures commission merchant (FCM) A futures trade facilitator who provides a one-stop service for all aspects of futures trading. Futures contract A futures contract is an exchange-traded agreement between a buyer and a seller to trade some commodity at a fixed price at a later date.

Gamma The sensitivity of the delta to changes in the underlying asset price. Gilt security Bond issued by the UK government. Good-till-canceled (GTC) order A limit order that stays open indefinitely until it is executed or closed by the trader who placed the order. Great Recession A commonly used term for 2007–9, when many countries suffered a decline in output and experienced large unemployment. Gross domestic product The market value of all goods and services produced in a year in a country. Hedge Protecting one’s investment against loss by making balancing or compensating contracts or transactions. Hedge fund A private actively managed investment fund that utilizes sophisticated strategies to generate returns for high-net-worth individuals. Hedge ratio The number of shares of the stock to trade for each option to remove all of the price risk from the resulting portfolio. Hedger Trader whose aim is to reduce preexisting risk by trading derivatives and other securities. High-frequency trading A computerized trading strategy pursuing fleeting profit opportunities by trading at the blink of an eye. Horizontal spread (time or calendar spread) A portfolio obtained by buying an option with a particular expiration date and simultaneously selling an option with a different maturity date, where the options (both calls or both puts) have the same underlying asset and the same strike price.

698

GLOSSARY

Hybrid debt (or structured note) A derivative security created by combining a bond and equity or a derivative (a forward, an option, or a swap).

Intracommodity spread (futures) A spread established by simultaneously buying and selling futures contracts on the same commodity but with different maturities.

Implied volatility An option’s underlying asset’s return volatility computed by equating the option’s market price to the option’s model price.

Intrinsic value (Asset) The present value of future cash flows, also known as fundamental value. (Option) The larger of the cash flow obtained by immediately exercising an option or zero.

Index (stock market) A measure computed by taking the average of a portfolio of stock prices. Index arbitrage An arbitrage trading strategy based on a discrepancy between a stock market index and a futures contract on the index. Index mutual fund (or stock index fund) A mutual fund that holds a basket of securities in the same proportion as an index it tries to replicate. Indexing A popular name for investing in index portfolios. Indication schedule (swap) A list of the various rates at which a swap dealer bank is willing to trade against sixmonth bbalibor. Inflation A rise in a general consumer price level. Initial margin (or performance margin) Before initiating a trade, a trader must keep this amount (in the form of cash or high-quality, low-risk securities) as collateral with the broker. Insurance A financial contract that requires the buyer to make periodic payments to an insurance company for protection against a loss event. Intercommodity spread (futures) A spread established by simultaneously buying and selling futures contracts on different commodities with the same or different delivery months.

Law of one price The condition that the same future cash flows, no matter how they are created, have the same market price. LEAPS (Long-Term Equity Anticipation Securities) Exchange-traded long maturity options that have maturities up to three years. Leverage The borrowing used to purchase a security or portfolio of securities. Liability A financial obligation that requires the holder to make payments at some future date. Libor (London Interbank Offered Rate) and libid An annualized interest rate at which a bank offers to lend surplus funds to another bank for a fixed time period in the London market. Libid (London Interbank Bid Rate) is a similar rate at which a bank bids funds. Limit order A trade given to a broker that must be filled at the stated or a better price or not traded at all. Libor index rate The daily rate determined by the Intercontinental Exchange by collecting libor quotes on Eurodollar deposits (or time deposits in nine other currencies) with maturities ranging from overnight to a year from a panel of major banks and computing a trimmed average.

Interest rate The rate of return on cash borrowed or lent.

Limit order book A dealer’s book (or an electronic record system) that records orders waiting to be filled.

International Money Market A division of the Chicago Mercantile Exchange (now part of the CME Group), where futures on foreign currencies trade.

Listing requirements Criteria that a company must satisfy to trade its stock on an exchange.

International Organization of Securities Commissions (IOSCO) An international cooperative body of securities regulators. International Swaps and Derivatives Association (ISDA; previously International Swap Dealers Association) An industry body consisting of financial institutions, businesses, governmental entities, and other end users that trade over-the-counter derivatives. In-the-money (option) An option having a positive value when immediately exercised.

Long The buyer of a security. Maintenance margin The minimum amount that a trader must keep in a margin account to keep a trading position open. Margin Collateral kept by a trader with a broker to cover the risk of losses on a transaction. Margin call A directive by a broker to increase the cash in a margin account when a trade position loses value and declines below the maintenance margin.

GLOSSARY

699

Marked-to-market (futures) A futures position is marked-to-market at the end of each trading day when gains and losses incurred on the futures contract are added to the trader’s margin account.

Modified duration Duration divided by one plus the bond’s yield.

Market close (or the closing call) The last minute of trading during which a closing price or a closing range is established.

Mortgage A loan that requires regular repayment of interest and principal where the loan is collateralized by a home or building.

Market corner A market in which a manipulator has control over the deliverable supply of a security or commodity so that short sellers are forced to pay the manipulator a high price to fulfill their contract obligations.

Municipal bond (muni) Bonds issued by state, local, or municipal governments whose holders typically enjoy a variety of tax-exemption benefits on interest earned.

Market imperfections (or market frictions) Impediments or costs to a trade such as transaction costs (brokerage commissions and bid–ask prices), trading restrictions, or taxes incurred when transacting in a security. Market maker Specially designated dealer on many exchanges who posts ask and bid prices and stands ready to trade at those prices. Market manipulation Trading securities using market power or other means to influence a securities price to one’s own profit advantage. Market order A trade that must be immediately transacted at the best available price. Market portfolio A portfolio that holds all the risky assets in a market in the same proportions as they exist in the economy. Market-value (or capitalization) weighted average A stock price index computed by multiplying each company’s share price by a weight that is proportionate to its market value (or capitalization) and then computing their average. Mark-to-model The valuation of assets and liabilities via the use of a model. Married put A portfolio obtained by simultaneously buying a put and its underlying asset. Martingale A stochastic process in which the expected value at a future date equals its current value. Martingale pricing A method of derivatives pricing that involves computing the expected discounted payoffs to the derivative using the pseudo-probabilities and discounting them backward through time with the riskless rate. Master Agreement (ISDA) A comprehensive document that consists of the legal conditions agreed to when trading over-the-counter derivatives.

Money market account An investment account that earns the risk-free rate each period.

Municipal swap A financial swap that has a state, local, or municipal government as one of the counterparties. Mutual fund Open-end investment company that takes new investments and allows investors to redeem their shares at any time. Naked trading strategy Option trade involving a single option with no corresponding position in the underlying asset. National Association of Securities Dealers (NASD) A self-regulatory organization that was created under the provisions of the US Securities Exchange Act of 1934 for regulating the US securities industry. National Association of Securities Dealers Automated Quotations (NASDAQ) Initially a computer bulletin board system that listed stock price quotes; evolved to become the world’s first electronic stock exchange. National Futures Association (NFA) An industry organization that does a host of self-regulatory, training, promotional, and other activities for futures contracts. Net asset value (NAV) A company’s total assets minus its total liabilities. Net present value Today’s value of a set of future cash flows, subtracting the initial cost. Noncommercial trader (futures) Trader other than a commercial trader; usually associated with speculation in futures contracts. Nondiversifiable risk Risk that cannot be eliminated through diversification in a portfolio of securities. Normal backwardation A forward or futures market in which the forward or futures price, respectively, is less than the expected future spot price.

700

GLOSSARY

Notional principal A principal amount that is used for computing cash flows in a derivatives transaction but never changes hands between the counterparties. Notional variable Variable such as interest rate, inflation rate, and security index that exists as a notion rather than as a traded asset. On-the-run issue vs. off-the-run issue Newly auctioned Treasury security vs. an issue that has been auctioned earlier. Open interest The total number of outstanding derivatives contracts. Open-end company Investment company that is open to additional investments. Opening call (futures) The initial time period for trading a futures contract. Open-outcry trading A traditional method of trading futures contracts in which the floor broker or dealer has to cry out the bids and asks in the trading pit so that others may participate.

Payout ratio The ratio of dividends paid on a stock to earnings per share. Physical delivery (forward and futures contracts) The process of closing out a forward or a futures contract by physically delivering the underlying asset or its ownership rights. Pink quote An electronic system that displays quotes from broker dealers for small stocks trading in the over-thecounter markets. Pit (ring) Polygonal area on an exchange floor where futures trading takes place. Plain vanilla derivatives Market jargon for the simplest derivatives contracts. Ponzi scheme An investment scheme that pays early investors’ returns from the cash coming from later investors. Portfolio insurance Synthetic put options created by trading the underlying stock and money market account using the option’s delta.

Option class The set of all call options or put options with a particular underlying asset.

Position limit The maximum number of futures (derivatives) contracts that a trader can hold on a particular side of the market.

Options Clearing Corporation (OCC) A clearinghouse for most exchange-traded options in the United States.

Position trader (trend follower) Speculator who maintains trading positions for longer than a day.

Options on futures (or futures options) Options that have a futures contract as the underlying asset.

Post (trading post) Specified places on an exchange’s trading floor where traders meet to trade.

Options series A set of options with identical terms and conditions.

Premium (option) The price of an option.

OTC Bulletin Board (OTCBB) A regulated quotation service that displays real-time quotes, last-sale prices, and volume information on over-the-counter equity securities. Out trade A trade that cannot be cleared by a clearinghouse because of the recording of incorrect information. Out-of-the-money (option) An option that generates a negative cash flow when immediately exercised. Over-the-counter (OTC) market Anywhere trading takes place other than an organized exchange.

Price bubble The difference between an asset’s market price and its fundamental value. Price discovery The process of discovering the price of a commodity through the interactions of buyers and sellers in a market. Price limit (daily; or the maximum daily price fluctuation) Maximum price change allowed during a trading day. Price-weighted average (stock index) A stock index computed by taking a simple average of the prices of the stocks composing the index.

Overwrite (option) Selling a call option after purchasing the underlying stock.

Primary dealer Big securities firms that actively bid in Treasury auctions and with whom the Fed buys and sells Treasuries to conduct open-market operations.

Par bond A bond for which the coupon rate is the same as its yield so that the bond’s price is the same as the bond’s face value.

Primary market The market in which a security is first sold to the public.

Par swap rate The swap rate initially set on a swap to give it zero initial value.

Program trading Trading that takes place using a computer program.

Principal Amount due on a bond at maturity.

GLOSSARY

Protective put A portfolio consisting of buying the put to protect the downside risk of the underlying asset that was previously purchased. Pseudo-probability (martingale probability or riskneutral probability) The risk-adjusted probability used in martingale pricing for computing derivatives prices. Put option A financial security that gives the right to sell a specified quantity of an underlying asset on or before a fixed future date by paying an exercise price. Putable bond A bond that gives the holder the option to end the bond at certain specific dates. Put–call parity A mathematical relation linking prices of a call, a put, a stock, and a bond. Random walk A stochastic process whose changes are independent and identically distributed random variables. Ratio spread An option trading strategy where the number of calls (or puts) purchased is different from the number of calls (or puts) sold.

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Savings and loans bank A special kind of bank in the United States that specializes in home mortgage lending. Scalper Speculator who trades many times a day with the hope of picking up small profits from each transaction. Scenario analysis (stress testing) A method for computing losses based on identifying possible future scenarios for the economy. Seat (in the context of a securities exchange) Membership on an exchange. Securities and Exchange Commission (SEC) The US federal government agency in charge of regulating equity and equity options markets. Securitization The process of combining many individual loans and debts into a collateral pool and selling claims against them to third-party investors. Security A financial contract (such as a stock, bond, or derivative) that gives its holder ownership rights over some future cash flows.

Real asset Asset including land, buildings, machines, or commodities, which have a physical form.

Self-financing trading strategy A trading strategy over multiple periods that requires no cash inflow or outflow at intermediate dates.

Reinsurance The practice of insurance companies buying insurance policies on the tail risks generated by the insurance policies they have issued.

Self-regulatory organization (SRO) An organization made up of members of an industry or a profession that exercises regulatory authority over it.

Repurchase agreement (“does a repo”) The borrowing of cash (and the loaning of securities) done by selling high-quality securities and then agreeing to purchase them back at a fixed date (often the next day) at a higher price.

Semiannual bond equivalent basis Computing interest over a half-year period under the assumption that the year has 365 days and the interest is computed over the actual number of days in the six-month period.

Residential mortgage–backed security (RMBS) An asset-backed security whose collateral pool consists of residential mortgage loans. Rho A measure of the change in an option value due a change in the risk-free interest rate. Risk premium The excess expected return per unit of risk on a security. Risk-neutral valuation A method for computing a derivative’s value by computing its expected payoffs using the pseudo-probabilities and discounting using the riskfree rate. Round lot A trading unit of stocks, usually based on one hundred shares. Round trip (commissions for trading futures) No commission charged when entering a position but full commission charged at the time of closing.

Separate trading of registered interests and principal of securities (STRIPS) Zero-coupon bonds that can be created by separating out different cash flows from a Treasury security. Settlement The last step of a security transaction where the buyer pays cash and gets the ownership rights over the security. Settlement date (forward rate agreement) The date on which a forward rate agreement’s reference rate is determined. Settlement price (futures) The last trading price or an appropriate price determined by an exchange’s settlement committee, which is used for daily settlement of the futures contract. Shareholder’s equity (equity) The assets owned minus the liabilities incurred by a company. Short Market jargon to mean the seller of a security.

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GLOSSARY

Short selling Borrowing and selling a security one does not own. Simple forward rate The forward rate expressed as an annualized simple interest rate. Special purpose vehicle (SPV) A legal entity set up for the purpose of issuing liabilities that are backed by a collection of assets in a collateral pool. Specialist A dealer-broker market maker whose job is to make an orderly market in securities assigned to him. Speculation The taking of risky trades in the hope of obtaining positive returns. Spin-off The creation of an independent company by selling shares or distributing as new shares an existing business or a division of a parent company. Spot price (or cash price) The price for an immediate transaction of a commodity or security. Spot rate The yield on the shortest-maturity default-free zero-coupon bond. Spread trading Trading that involves buying one or more securities and selling similar securities with the aim of generating arbitrage profits as the price differences converge.

Swap facilitator (swap bank) Financial intermediary who acts as a broker or performs the role of a dealer in the swap market. Swap issuer (initiator) A party that initiates a financial swap by seeking out counterparties to complete the transaction. Swap rate The predetermined interest rate that a financial swap counterparty pays in exchange for a well-known reference rate such as the bbalibor. This rate gives the swap a zero value at initiation. Swaption An option that has an interest rate swap as the underlying asset. Synthetic index A portfolio created for the purpose of replicating the returns on an index. Synthetic option A portfolio of traded securities that replicates an option’s payoff and consequently must have the same price as the traded option to avoid arbitrage. Tail risk The risk of occurrence of infrequent events that lie on the tail of a probability distribution.

Stock A limited liability asset that gives the investor ownership rights over the residual value of a company.

Technical analysis A security analysis technique that involves looking at price patterns and related measures to predict whether a stock’s price is overvalued or undervalued.

Stock price evolution A model that describes how a stock price evolves randomly across time.

Tenor (maturity or term of a swap) Time duration of a financial swap.

Stop-loss order A limit sell order that is placed below current market prices or a limit buy order that is placed above current prices.

Term structure of interest rates The graph of yields or forward rates versus the time to maturity.

Storage costs Costs that are incurred for storing a physical commodity in the context of the cost-of-carry model. Straddle (called put-to-call strategy in London) An option trading strategy that involves simultaneously buying (or selling) a call and a put on the same underlying asset with the same strike price and expiration date. Strangle An option trading strategy that involves simultaneously buying (or selling) a call and a put on the same underlying asset with the same expiration date but different strike prices. Subprime lending (also known as B-paper, nearprime, or second chance lending) The practice of making loans to borrowers who do not qualify for the best market interest rates (the prime market) because of a poor credit history. Swap (financial swap) An agreement between two counterparties to exchange a series of (usually) semiannual cash flows over the life of the contract. Swap buyer Counterparty to a (plain vanilla) financial swap that pays fixed and receives the floating rate.

Theta A measure of the change in an option’s value given a decrease in the option’s time to maturity. Tick size (minimum price fluctuation) The minimum price move for an exchange-traded security. Ticker (trading) symbol The abbreviation that identifies an exchange-traded security. Ticker tape The running paper tape or electronic panel displaying a variety of prices, quotes, and relevant news to subscribers located both in and off the exchange. Time spread (calendar spread) A spread established by simultaneously buying and selling futures contracts on commodities with different delivery months. Time value (option) An option’s premium (or price) minus its intrinsic value. Total return index An index that assumes that all disbursements from the companies in the index get reinvested in a hypothetical index portfolio. Total return swap A swap where one counterparty pays the total return on a price index in exchange for a fixed or floating interest rate payment.

GLOSSARY

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Trading at settlement (or TAS) Special futures contracts in which the counterparties decide at the time of trading that the contract price will be the day’s settlement price plus or minus an agreed differential.

a particular strike price while simultaneously selling an option with a different strike price, where the options (both calls or both puts) have the same underlying asset and expire on the same date.

Trading volume (or volume) The total number of contracts or securities traded during a trading day.

VIX (CBOE Volatility index) A measure of the market’s near-term volatility as implied by the Standard and Poor’s 500 stock index option prices.

Treasury bill (T-bill) US government debt sold in the form of a zero-coupon bond that has a maximum maturity of one year. Treasury inflation protected security (TIPS) Treasury security that adjusts for inflation and pays a guaranteed real rate of return to the investors. Treasury note and bond (T-note and T-bond) Treasury securities that make semiannual coupon payments with original maturity of more than one year to up to ten years in the case of a Treasury note and more than ten years in the case of a Treasury bond. Treasury security (Treasuries) Debt issued by the US federal government. Treasury–Eurodollar (TED) spread The difference between the bbalibor rate for US dollars and Treasury rates of equivalent maturities. Underlying (for a derivative) A reference asset or a notional variable whose price or value determines the derivative’s payoff. Uniform price auction An auction in which all successful bidders pay the same price (the highest losing bid or the lowest winning bid). Upstairs market A market away from the trading floor in which large-sized trades are negotiated, with or without the help of brokers. Value-at-risk (VaR) The dollar amount that the firm’s losses will exceed with a given probability (such as 1 percent or 5 percent) over a prespecified time interval. Variation margin An amount equal to the change in the futures settlement price from the previous trading day that is credited or debited to a trader’s account. Vega A measure of the change in an option’s value given a change in the underlying asset’s return volatility. Vertical spread (money, perpendicular, or price spread) A portfolio consisting of buying an option with

Wall Street An expression for investment banking companies, many of which have offices near Wall Street in New York’s financial district. Warehousing swap A risk management technique used by swap providers in which they enter into different swaps on both the buy and sell side of the market and hedge the residual risk. Wash sale A sell and buy transaction for the same commodity initiated without the intent to make a bona fide transaction. When-issued market A specialized forward market for Treasury securities that opens before the actual securities are auctioned. Wild card play (T-bond futures) An embedded option available to the seller of Treasury bond futures to issue the notice of intention to deliver for a few hours after the market closes. Writer (option) Seller of an option. Yield (yield to maturity or internal rate of return) The interest rate that discounts a bond’s cash flows to equal its current price. Yield curve A graph of bond yields as a function of maturity. Zero-cost collar A portfolio that combines a long position in the underlying asset with a long put and a short call so that the resultant portfolio has truncated payoffs on both up and down sides. Zero-coupon bond A bond that pays no interest but is sold at a discount from the principal. Zero-sum game A game in which one trader’s gain is the other’s loss.

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Mehrling, Perry, 2005. Fischer Black and the Revolutionary Idea of Finance. John Wiley & Sons, Hoboken, N.J. Merton, Robert C., 1973a. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science 4(1), 141–83. Merton, Robert C., 1973b. “An Intertemporal Capital Asset Pricing Model.” Econometrica 41(5), 867–87. Merton, Robert C., 1974. “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29(2), 449–70. Merton, Robert C., 1977. “An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees: An Application of Modern Option Pricing Theory.” Journal of Banking and Finance 1(1), 3–11. Merton, Robert C., and Myron S. Scholes, 1995. “Fischer Black.” Journal of Finance 50(5), 1359–70. Miller, Merton H., 1986. “Financial Innovation: The Last Twenty Years and the Next.” Journal of Financial and Quantitative Analysis 21(4), 459–71. Miller, Merton H., 1997. Merton Miller on Derivatives. John Wiley, New York.

Pan, Jun, 2002. “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study.” Journal of Financial Economics 63(1), 3–50. Park, Hun Y., and Andrew H. Chen, 1985. “Differences Between Futures and Forward Prices: A Further Investigation of Marking to Market Effects.” Journal of Futures Markets 5(1), 77–88. Park, James A. 1799. A System of the Law of Marine Insurances. Macmillan and Co., London. Park, Yoon S. 1984. “Currency Swaps as a Long-Term International Financing Technique.” Journal of International Business Studies 15(3), 47–54. Paulson, Jr., Henry M., 2010. On the Brink: Inside the Race to Stop the Collapse of the Global Financial System. Business Plus, New York. Qin, Wang X. and Nick Ronalds, 2005. “China: The Fall and Rise of Chinese Futures, 1990–2005.” Futures Industry Inside May/June,www.futuresindustry.org/fimagazine-home.asp?a=1038.

Miller, Merton H., and Franco Modigliani, 1961. “Dividend Policy, Growth, and the Valuation of Shares.” Journal of Business 34(4), 411–33.

Rebonato, Riccardo, 2002. Modern Pricing of Interest Rate Derivatives: The LIBOR Market Model and Beyond. Princeton University Press, Princeton, N.J.

Miltersen, Kristian R., Klaus Sandmann, and Dieter Sondermann, 1997. “Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates.” Journal of Finance 52(1), 409–30.

Rendleman, Richard J., and B. Bartter, 1979. “Two State Option Pricing.” Journal of Finance 34(5), 1092–110.

Modigliani, Franco, and Merton H. Miller, 1958. “The Cost of Capital, Corporate Finance, and the Theory of Investment.” American Economic Review 48(3), 261–97. Modigliani, Franco, and Richard Sutch, 1966. “Innovations in Interest Rate Policy.” American Economic Review 56(1/2), 178–97. Mood, Alexander M., Franklin A. Graybill, and Duance C. Boes, 1974. Introduction to the Theory of Statistics. McGraw-Hill Higher Education, New York. Mossin, Jan, 1966. “Equilibrium in a Capital Asset Market.” Econometrica 35(4), 768–83. Nelson, Samuel A., 1904. The A B C of Options and Arbitrage. S. A. Nelson, New York. Ofek, Eli, Matthew Richardson, and Robert F. Whitelaw, 2004. “Limited Arbitrage and Short Sales Restrictions: Evidence from the Options Markets.” Journal of Financial Economics 74(2), 305–42. Overby, Brian, 2007. The Options Playbook. TradeKing.

Rendleman, Richard J., and Christopher E. Carabini, 1979. “The Efficiency of the Treasury Bill Futures Market.” Journal of Finance 34(4), 895–914. Ross, Stephen A., Randolph W. Westerfield, and Jeffrey Jaffe, 2004. Corporate Finance. 7th ed. McGrawHill/Irwin, New York. Samuelson, Paul A., 1965a. “Rational Theory of Warrant Pricing.” Industrial Management Review 6, 13–39. Samuelson, Paul A., 1965b. “Proof That Properly Anticipated Prices Fluctuate Randomly.” Industrial Management Review 6, 41–50. Samuelson, Paul A., and Robert C. Merton, 1969. “A Complete Model of Warrant Pricing That Maximizes Utility.” Industrial Management Review 10(2), 17–46. Sandmann, Klaus, Dieter Sondermann, and Kristian R. Miltersen, 1995. “Closed Form Term Structure Derivatives in a Heath Jarrow Morton Model with LogNormal Annually Compounded Interest Rates.” Proceedings of the Seventh Annual European Research Symposium, Bonn, September.

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Notation

A(t)

b(t)

Value of a money market account that starts with an investment of $1 and earns the default-free spot rate each period (chapters 1–25) or the value of a firm’s assets (chapter 26). Basis (≡ spot price minus futures price) at time t.

B(t,T) The price at time t of a zero-coupon bond paying $1 at time T (sometimes shortened as B(t) or B). BE

Zero-coupon bond price in a foreign country (in “European terms”).

F(t)

Futures price at time t.

g

Storage cost (annual rate).

G

Storage cost (lump sum).

h

The minimum-variance futures hedge ratio.

i

Simple (“domestic” or “dollar”) interest rate per year; iE is (“European” or “foreign”) interest rate per year.

I

Index. Futures interest rate.

C

Cash flow or a bond’s coupon.

ifut

c

European call price or caplet price (chapters 1–25) or the market clearing CDS rate (chapter 26).

iS

Swap rate.

i(t,T)

cA

American call price.

Simple forward rate considered at time t, which becomes effective at time T.

corr

Correlation coefficient.

log

Natural logarithm.

cov

Covariance.

L

Principal or par value of a bond.

𝛿

Dividend yield (chapters 1–24), or the time interval for computing a simple forward rate (chapter 25), or the recovery rate (chapter 26).

L(t)

Value of a debt issue at time t.

LN

Notional principal.

𝜆

div

Dollar dividend.

dur

Duration of a bond.

Probability of default per unit time (in reducedform model). 𝜆̃ is the risk neutral default probability.

durM

Modified duration of a bond.

𝜇

Δ

(Delta) Change in a variable’s value.

Expected return (chapters 1–24) or expected percentage change in the simple forward rate (chapter 25).

e

Exponential function.

N(x)

E(t)

A firm’s equity value at time t.

Cumulative standard normal distribution function with mean 0 and standard deviation 1.

E( . )

Computing expectation of the expression inside parentheses.

E𝜋 ( . ) Computing expectation of the expression inside parentheses using the pseudo-probabilities denoted by 𝜋. fFRA

Forward rate agreement rate.

f (t,T) Forward rate contracted at time t that is effective over the time period [T,T + 1]. F(t,T) Forward price at time t for a contract maturing at time T (sometimes shortened as F). FA

Forward price of a non-US currency in American or dollar terms.

FE

Forward price of a non-US currency in European terms or in terms of a currency other than the US dollar.

N ′ (x) Standard normal density function. p

European put price (chapters 1–20) or floorlet price (chapters 21–26).

pA

American put price.

P

Bond price.

Prob

Probability.

Prob𝜋 Probability that a random variable takes on certain values when 𝜋 is the underlying probability. 𝜙

Term for risk adjustment in pseudo-probability.

𝜋

Pseudo-probability that a stock price moves up.

𝜋(t,T) Pseudo-probability that a T-maturity zero-coupon bond moves up at time t. PV

Present value.

q

Actual probability that a stock price moves up.

710

NOTATION

711

q(t)

Actual probability that the term structure of zerocoupon bond prices moves up at time t.

SCDIV Stock price cum-dividend.

r

Continuously compounded interest rate per year. rE Foreign interest rate (interest rate in “European” terms).

sd

Standard deviation.

𝜎

Standard deviation (volatility) of stock return per year.

𝜃

Stock’s risk premium.

𝜐

Average forward rate volatility.

U, D

Up and down factor, respectively, in a singleperiod or multiperiod binomial option pricing model.

R(0,T) Dollar return at time T from investing $1 in riskfree bonds today (chapters 1–20). R(t)

Default-free spot rate of interest declared at time t that applies to the time period [t,t + 1].

RE

“Dollar return” concept applied to investment in foreign currencies (in European terms or in terms of a currency other than the US dollar).

Ref

Reference rate (bbalibor or euribor in most cases) announced on some future date.

S(t)

Stock price or spot price of a commodity at time t.

SA

Spot exchange rate in American terms.

SE

Spot exchange rate in European terms.

SXDIV Stock price ex-dividend.

u(t,T) Up and down factor, respectively, in an HJM d(t,T) multiperiod binomial interest rate option pricing model. var

Variance.

VaR𝛼

Value-at-risk for an 𝛼 percent loss probability.

y

The convenience yield per year (chapter 12), or the yield on a coupon-bond (chapters 21–26).

Y

Lump sum convenience yield.

Additional Sources and Websites

Journals Academic journals are the best source for research on derivatives and risk management. They include the following.

General Finance Journals ■

Annual Review of Financial Economics, www.annualreviews.org/journal/financial

■

Finance and Stochastics, www.springer.com/mathematics/quantitative+finance/ journal/780

■

Financial Analysts Journal, www.cfapubs.org/loi/faj

■

Financial Management, https://www.fma.org/financial-management

■

Journal of Banking & Finance, www.journals.elsevier.com/journal-of-bankingand-finance/

■

Journal of Finance, www.afajof.org

■

Journal of Financial and Quantitative Analysis, www.jfqa.org

■

Journal of Financial Economics, http://jfe.rochester.edu

■

Mathematical Finance, https://onlinelibrary.wiley.com/journal/14679965

■

Review of Financial Studies, http://rfs.oxfordjournals.org

Derivatives and Risk Management Journals ■

Journal of Derivatives, https://jod.iijournals.com/

■

Journal of Fixed Income, https://jfi.iijournals.com/

■

Journal of Futures Markets, http://onlinelibrary.wiley.com/journal/10.1002/%28 ISSN%291096-9934

■

Journal of Risk, www. thejournalofrisk.com

■

Review of Derivatives Research, www.springer.com/business+%26+management/finance/ journal/11147

■

Risk Magazine (publishes mathematical finance articles), www.risk.net/riskmagazine

Economics Journals with Occasional Articles on Derivatives and Risk Management ■

American Economic Review, www.aeaweb.org/aer/index.php

■

American Journal of Agricultural Economics, http://ajae.oxfordjournals.org/

■

Econometrica, www.wiley.com/WileyCDA/WileyTitle/productCd-ECTA.html

■

Journal of Economic Perspectives, www.aeaweb.org/jep/

ADDITIONAL SOURCES AND WEBSITES

■

Journal of Economic Surveys, www.wiley.com/WileyCDA/WileyTitle/productCdJOES.html

■

Journal of Political Economy, www.press.uchicago.edu/ucp/journals/journal/jpe.html

■

Review of Economics and Statistics, www.mitpressjournals.org/loi/rest

Mathematics, Statistics, and Management Journals with Occasional Articles on Derivatives and Risk Management ■

Annals of Applied Probability, www.imstat.org/aap/

■

Journal of the American Statistical Association, www.amstat.org/publications/jasa.cfm

■

Stochastic Processes and Their Applications, www.journals.elsevier.com/stochasticprocesses-and-their-applications/

Newspapers and Magazines Business newspapers and magazines frequently write about derivatives and risk management. ■

Barron’s, http://online.barrons.com

■

Bloomberg, www.bloomberg.com

■

The Economist, www.economist.com

■

Financial Times, www.ft.com

■

Futures and Options World, www.fow.com

■

Harvard Business Review, http://hbr.org/magazine

■

Investor’s Business Daily, www.investors.com

■

New York Times, www.nytimes.com

■

Wall Street Journal, http://online.wsj.com

Government Agencies Nowadays, many government agencies have websites; examples follow. ■

US Department of the Treasury, www.treasury.gov

■

Bank for International Settlements, www.bis.org

■

Board of Governors of the Federal Reserve System, www.federalreserve.gov

■

Federal Reserve Bank of New York, www.newyorkfed.org

■

International Monetary Fund (IMF), www.imf.org

■

US Securities and Exchange Commission, www.sec.gov

■

US Department of the Treasury, www.treasury.gov

■

World Bank Group, www.worldbank.org

713

714

ADDITIONAL SOURCES AND WEBSITES

Industrial Associations and Bodies, Self-Regulatory Organizations Many industry associations and bodies as well as self-regulatory organizations disseminate information through websites. ■

CFA Institute, www.cfainstitute.org

■

Financial Accounting Standards Board (FASB), www.fasb.org

■

Financial Industry Regulatory Authority (FINRA), www.finra.org

■

Global Association of Risk Professionals (GARP), www.garp.org

■

International Association of Financial Engineers (IAFE), http://iafe.org

■

International Swaps and Derivatives Association Inc. (ISDA), www2.isda.org

■

National Futures Association (NFA), www.nfa.futures.org

■

Securities Industry and Financial Markets Association (SIFMA), www.sifma.org

Exchanges Most exchanges have websites that include a description of their business, a historic timeline, contract specifications, price and volume information, and educational and promotional material. ■

BM&FBovespa, http://www.b3.com.br/en_us/

■

Chicago Board Options Exchange (CBOE), www.cboe.com

■

CME Group, www.cmegroup.com

■

Eurex (includes ISE), www.eurexchange.com and www.ise.com

■

Korea Exchanges (KRX), http://eng.krx.co.kr

■

Multi Commodity Exchange of India (MCX), www.mcxindia.com

■

NASDAQ OMX, www.nasdaqomx.com

■

National Stock Exchange of India (NSE), www.nseindia.com

■

NYSE Euronext, https://nyse.nyx.com

■

Russian Trading Systems Stock Exchange (RTS), www.rts.ru/en/

Financial Institutions Predictions on market directions and economic conditions, advice on trading strategies, and risk management–related issues are found in bulletins, newsletters, and research reports from a wide variety of sources, including banks, brokerage firms, commodity trading advisors, credit rating agencies, finance companies (including mutual funds), and hedge fund managers. These can be found on their respective websites.

Books on Derivatives and Risk Management

Following is a representative list of books on derivatives and risk management. Abu-Mostafa, Yaser S., Blake LeBaron, Andrew W. Lo, and Andreas S. Weigend (eds.), 2000, Computational Finance 1999.

Chaplin, Geoff, 2010, Credit Derivatives: Trading, Investing, and Risk Management.

Acharya, Viral V., Thomas F. Cooley, Matthew P. Richardson, and Ingo Walter (eds.), 2010, Regulating Wall Street: The Dodd-Frank Act and the New Architecture of Global Finance.

Chriss, Neil A., 1997, Black–Scholes and Beyond: Option Pricing Models.

Andersen, Leif B. G., and Vladimir V. Piterbarg, 2010, Interest Rate Modeling, 3 volumes. Augen, Jeffrey, 2011, Microsoft Excel for Stock and Option Traders: Build Your Own Analytical Tools for Higher Returns. Back, Kerry, 2010, A Course in Derivative Securities: Introduction to Theory and Computation. Barker, Philip, 2007, Java Methods for Financial Engineering: Applications in Finance and Investment. Baxter, Martin, and Andrew Rennie, 1996, Financial Calculus: An Introduction to Derivatives Pricing. Bielecki, Tomasz, Damiano Brigo, and Frederic Patras, 2011, Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings, and Liquidity. Bouzoubaa, Mohamed, and Adel Osseiran, 2010, Exotic Options and Hybrids: A Guide to Structuring, Pricing and Trading. Boyle, Phelim, and Feidhlim Boyle, 2001, Derivatives: The Tools That Changed Finance. Brandimarte, Paolo, 2006, Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, 2nd edition. Brigo, Damiano, and Fabio Mercurio, 2006, Interest Rate Models: Theory and Practice: With Smile, Inflation and Credit. Butler, Cormac, 2009, Accounting for Financial Instruments.

Choudhry, Moorad, 2010, The Repo Handbook, 2nd edition.

Culp, Christopher L., 2006, Structured Finance and Insurance: The ART of Managing Capital and Risk, 2nd edition. Das, Satyajit, 2005, Credit Derivatives: CDOs and Structured Credit Products, 3rd edition. DeRosa, David F., 1998, Currency Derivatives: Pricing Theory, Exotic Options, and Hedging Applications. Dixit, Avinash K., and Robert S. Pindyck, 1994, Investment under Uncertainty. Duffie, Darrell, 1989, Futures Markets. Duffie, Darrell, and Kenneth J. Singleton, 2003, Credit Risk: Pricing, Measurement, and Management. Edwards, Franklin R., and Cindy W. Ma, 1992, Futures and Options. Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch, 1997, Modelling Extremal Events for Insurance and Finance. Fabozzi, Frank J. (ed.), 2005, The Handbook of Mortgage Backed Securities, 6th edition. Fabozzi, Frank J. (ed.), 2011, The Handbook of Fixed Income Securities, 8th edition. Fabozzi, Frank J., Roland Fuss, and Dieter G. Kaiser, 2008, The Handbook of Commodity Investing. Fabozzi, Frank J., and Vinod Kothari, 2008, Introduction to Securitization.

Chacko, George, Vincent Dessain, Peter Hecht, and Anders Sjoman, 2006, Financial Instruments and Markets: A Casebook.

Figlewski, Stephen, William L. Silber, and Marti G. Subrahmanyan, 1990, Financial Options: From Theory to Practice.

Chacko, George, Anders Sjöman, Hideto Motohashi, and Vincent Dessain, 2006, Credit Derivatives: A Primer on Credit Risk, Modeling, and Instruments.

Fouque, Jean-P., George Papanicolaou, Ronnie Sircar, and Knut Sølna, 2011, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives.

Flavell, Richard, 2010, Swaps and Other Derivatives.

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BOOKS ON DERIVATIVES AND RISK MANAGEMENT

Froot, Kenneth A. (ed.), 1999, The Financing of Catastrophe Risk. Geman, Hélyette, 2005, Commodities and Commodity Derivatives: Modelling and Pricing for Agriculturals, Metals and Energy. Glasserman, Paul, 2003, Monte Carlo Methods in Financial Engineering. Haug, Espen G., 2006, The Complete Guide to Option Pricing Formulas, 2nd edition. Ho, Thomas S. Y., and Sang-Bin Lee, 2004, The Oxford Guide to Financial Modeling: Applications for Capital Markets, Corporate Finance, Risk Management, and Financial Institutions. Hughston, Lane P. (ed.), 1997, Vasicek and Beyond: Approaches to Building and Applying Interest Rate Models. Hughston, Lane P. (ed.), 2000, The New Interest Rate Models: Recent Developments in the Theory and Application of Yield Curve Dynamics.

McLaughlin, Robert M., 1998, Over-the-Counter Derivative Products: A Guide to Legal Risk Management and Documentation. McMillan, Lawrence G., 2012, Options as a Strategic Investment, 5th edition. McNeil, Alexander J., Rüdiger Frey, and Paul Embrechts, 2005, Quantitative Risk Management: Concepts, Techniques, and Tools. Miller, Merton H., 1997, Merton Miller on Derivatives. Natenberg, Sheldon, 1995, Option Volatility and Pricing: Advanced Trading Strategies and Techniques, 2nd edition. Pachamanova, Dessislava, and Frank J. Fabozzi, 2010, Simulation and Optimization in Finance + Website: Modeling with MATLAB, @Risk, or VBA. Pearson, Neil D., 2002, Risk Budgeting: Portfolio Problem Solving with Value-at-Risk. Rebonato, Riccardo, 1999, Volatility and Correlation: In the Pricing of Equity, FX and Interest-Rate Options.

Hull, John C., 2011, Options, Futures, and Other Derivatives, 8th edition.

Rebonato, Riccardo, 2002, Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond.

Jarrow, Robert A. (ed.), 1998, Volatility: New Estimation Techniques for Pricing Derivatives.

Rendleman, Richard J., 2002, Applied Derivatives: Options, Futures and Swaps.

Jarrow, Robert A., 2002, Modeling Fixed Income Securities and Interest Rate Options, 2nd edition.

Ritchken, Peter, 1996, Derivative Markets: Theory, Strategy, and Applications.

Jarrow, Robert A., 2008, Financial Derivatives Pricing: Selected Works of Robert Jarrow.

Ronn, Ehud I., 2003, Real Options and Energy Management: Using Options Methodology to Enhance Capital Budgeting Decisions.

Jarrow, Robert A., and Andrew Rudd, 1983, Option Pricing. Jarrow, Robert A., and Stuart M. Turnbull, 2000, Derivative Securities, 2nd edition. Jha, Siddhartha, 2011, Interest Rate Markets: A Practical Approach to Fixed Income. Jorion, Philippe, 2006, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edition. Kat, Harry M., 2001, Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured Notes. Lindsey, Richard R., and Barry Schachter, 2009, How I Became a Quant: Insights from 25 of Wall Street’s Elite. Lipton, Alexander, and Andrew Rennie, 2011, The Oxford Handbook of Credit Derivatives. Mason, Scott P., Robert C. Merton, Andre F. Perold, and Peter Tufano, 1995, Cases in Financial Engineering: Applied Studies of Financial Innovation. McDonald, Robert L., 2009, Derivatives Markets, 3rd edition.

Saunders, Anthony, and Linda Allen, 2010, Credit Risk Management In and Out of the Financial Crisis: New Approaches to Value at Risk and Other Paradigms, 3rd edition. Schönbucher, Philipp J., 2003, Credit Derivatives Pricing Models: Model, Pricing and Implementation. Shiller, Robert J., 2004, The New Financial Order: Risk in the 21st Century. Shimko, David C. (ed.), 2004, Credit Risk: Models and Management, 2nd edition. Shreve, Steve E., 2005, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model; Stochastic Calculus for Finance II: Continuous-Time Models. Siegel, Daniel R., and Diane F. Siegel, 1990, Futures Markets. Smithson, Charles W., 1998, Managing Financial Risk: A Guide to Derivative Products, Financial Engineering, and Value Maximization, 3rd edition.

BOOKS ON DERIVATIVES AND RISK MANAGEMENT

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Stigum, Marcia L., and Anthony Crescenzi, 2007, Stigum’s Money Market, 4th edition.

Van Deventer, Donald R., and Kenji Imai, 2003, Credit Risk Models and the Basel Accords.

Sundaresan, Suresh, 2013, Fixed Income Markets and Their Derivatives, 4th edition.

Van Deventer, Donald R., Kenji Imai, and Mark Mesler, 2013, Advanced Financial Risk Management: Tools and Techniques for Integrated Credit Risk and Interest Rate Risk Managements, 2nd edition.

Tavakoli, Janet M., 2008, Structured Finance and Collateralized Debt Obligations: New Developments in Cash and Synthetic Securitization, 2nd edition. Tuckman, Bruce, and Angel Serrat, 2011, Fixed Income Securities: Tools for Today’s Markets, 3rd edition. Van Deventer, Donald R., and Kenji Imai, 2013, Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance and Investment Management.

Veronesi, Pietro, 2010, Fixed Income Securities: Valuation, Risk, and Risk Management.

Name Index

Alexander, Gordon J., 67 Alletzhauser, Albert J., 155 Amemiya, Takeshi, 281n6 Amin, Kaushik I., 296 Arrow, Kenneth, 8 Artzner, Philippe, 667 Bachelier, Louis, 369, 370 Bailey, Jeffery V., 67 Bank, Peter, 440 Barings, Francis, 260n1 Bartter, Brit J., 369, 370, 371, 407 Bakshi, Gurdip, 479 Baum, Dietmar, 440 Baxter, Martin W., 371n1, 564 Bernstein, Peter L., 656 Black, Fischer S., 8, 12, 312, 368, 370, 383, 423, 424, 429, 620–22, 647 Blanchard, Olivier, 20 Box, George E. P., 679 Brace, Alan, 565, 622, 648 Brealey, Richard, 507 Buffett, Warren, 10, 204, 312, 356 Çetin, Umut, 677 Chatterjea, Arkadev, 311, 678 Coase, Ronald, 7–9 Cox, David R., 453 Cox, John C., 369, 370, 371, 407, 504 Culbertson, John M., 504 Debreu, Gerard, 8 DiNardo, John, 281n6 Dixit, Avinash K., 673 Draper, Norman R., 679 Drew, Daniel, 314 Duffie, Darrell, 94, 272, 448 Dunn, Kenneth B., 504 Easterbrook, Frank H., 203 Edelman, Asher B., 114 Edwards, Franklin R., 83 Ehrlich, Paul R., 168, 168–69 Eisenberg, Larry, 473 Engle, Robert F., 312, 480 Evans, George W., 476n3

Fama, Eugene, 423 Feynman, Richard, 368 Fischel, Daniel R., 203 Fisher, Irving M., 504 Fisk, James, 202 Follmer, Holmer, 667 Froot, Kenneth A., 259 Gastineau, Gary L., 134 Gelderblom, Oscar, 291 Glasserman, Paul, 94, 448 Gould, Jay, 202 Greenbaum, Stuart I., 540 Greenspan, Alan, 10, 295 Gross, David, 10 Harrison, J. Michael, 370, 371 Harte, John, 169 Heath, David C., 131n1, 360n3, 565, 621 Hicks, John, 8, 504 Higgins, Leonard R., 337 Ho, Thomas S. Y., 595 Holden, Craig W., 490 Holdren, John, 169 Honkapohja, Seppo, 476n3 Hughston, Lane P., 565n2 James, William, 311n2 Jarrow, Robert A., 131n1, 185, 215, 244n2, 272, 296, 311, 357n2, 360n3, 361, 369–71, 375, 384, 407, 425, 428, 440, 449, 457, 473, 504, 565, 584, 593, 606, 607, 614, 615, 621, 626, 636, 647, 649, 667–69, 673, 675, 676, 678, 679 Jegadeesh, Narasimhan, 204 Johnson, R. Stafford, 540 Johnston, Jack, 281n6 Jones, E. Philip, 672 Jonker, Joost, 291 Jordan, Bradford D., 204 Jordan, Susan D., 204 Keynes, John Maynard, 8, 188 Knoll, Michael, 342

Kreps, David M., 370, 371 Kritzman, Mark P., 134 Lamont, Owen A., 343 Lee, Sang-Bin, 565n2, 595 Leeson, Nick, 260n1 Leland, Hayne, 672 Lintner, John, 8 Lipschutz, Bill, 311 Lutz, Friedrich A., 504 Lynch, Peter, 10 Ma, Cindy W., 83 Macaulay, Frederick R., 507 MacKenzie, Donald, 305 Madoff, Bernard, 201 Malkiel, Burton G., 291, 305 Malthus, Thomas, 168 Markowitz, Harry, 8 Mellon, Andrew William, 23 Meriwether, John, 677 Merton, Robert C., 8, 12, 368, 370, 371, 383, 423, 426, 669, 670 Miller, Hilton D., 453 Miller, Merton, 7, 8, 10, 342 Millo, Yuval, 305 Miltersen, Kristian R., 565, 622, 648 Modigliani, Franco, 8, 504 Mood, Alexander M., 453 Morgenbesser, Sidney, 320 Morton, Andrew J., 565, 621 Mossin, Jan, 8 Myers, Stewart, 507 Nelson, Charles R., 474n2 Nelson, Samuel A., 428, 463 Nocera, Joe, 656 Oppenheimer, Ernest, 196 Overby, Brian, 305 Pan, Jun, 479 Pindyck, Robert S., 673 Pliska, Stanley R., 370, 371 Ponzi, Charles, 201 Protter, Philip E., 357n2, 425

718

NAME INDEX

Qin, Wang X., 86 Quandt, Richard E., 291 Rebonato, Riccardo, 636 Rendleman, Richard J., 369, 370, 371, 407 Rennie, Andrew J. O., 371n1, 564 Ronalds, Nick, 86 Rosenberg, Joshua V., 480 Ross, David J., 203 Ross, Stephen A., 369, 672 Rubinstein, Mark, 369, 423 Rudd, Andrew, 369, 370, 371, 407 Sage, Russell, 292, 337 Samuelson, Paul A., 8, 370, 423, 426

Sandmann, Klaus, 565, 622, 648 Sargent, Thomas, J., 476n3 Sarnoff, Paul, 292 Schied, Alexander, 667 Scholes, Myron S., 8, 12, 368, 370, 383, 423, 429 Schwager, Jack, 311n1 Sharpe, William F., 8, 67, 370, 371 Shimbo, Kazuhiro, 425 Singleton, Kenneth J., 504 Sowell, Thomas, 268 Sullivan, Joseph, 306 Summers, Lawrence H., 289, 312 Sundaresan, Suresh, 504 Sutch, Richard, 504

719

Thakor, Anjan V., 540 Thaler, Richard H., 343 Thales of Miletus, 292 Tobin, James, 8 Turnbull, Stuart M., 185, 215, 272, 370, 371, 407, 669, 673 Varian, Hal R., 193 Vasicek, Oldrich A., 371, 565, 595 Weatherstone, Dennis, 662 Wessel, David, 5 West, Mark D., 155 Wilmott, Paul, 94, 448 Zhu Rongji, 289, 312

Subject Index

ABS CDSs (asset-backed security credit default swaps), 148 ABSs, see Asset-backed securities Active management, duration, 512 Actual probabilities and binomial HJM model, 576 and binomial options pricing model, 382–83, 399 and BSM model, 433–34, 457–58 and Eurodollar futures rates, 609–10 and HJM libor model, 637 pseudo- and, 382–83, 385–87, 433–34, 457–58, 576, 638 and risk premium, 385–88 Actuaries, 327 Adjustable-rate mortgage (ARM), 498 Airline fuel costs, hedging of, 146, 147, 260, 261 Airline miles, arbitrage with, 116 Alternative trading systems (ATSs), 61–62 American options arbitrage opportunities with, 353–54 binomial pricing model for, 410–12 Black–Scholes–Merton model for, 447–48 early exercise of, 357–62 exercise of, 94 HJM libor model for, 647 price bounds for, 99–100, 104, 346–48, 352 profit diagrams for, 313 put–call parity for, 344–46 value of European vs., 347 American Stock Exchange, 295 American terms, currency quotes, 138 Amortized swaps, 549 Amsterdam, Netherlands, 157 Amsterdam Stock Exchange, 295 Annual Percentage Rate (APR), 25 Annualized rate of return, 24–25, 45, 46 Annualized volatility, 443 Appreciation, US dollar, 138 APR (Annual Percentage Rate), 25 APs (associated persons), 173

Arbitrage (arb), 112–27; see also No-arbitrage principle and cost-of-carry model, 216 defined, 6 examples of, 116 index, 121–23 for indexes with dividends, 238 and market efficiency, 118–19 tax (regulatory), 342 in trading strategies, 119–23 Arbitrage-free conditions binomial trees under, 378, 573 futures prices under, 608–9 futures rates under, 609–10 for single-period binomial HJM model, 570–73 for two-period binomial HJM model, 597–99 Arbitrage opportunities, 63, 113–16 with American puts, 353 and binomial HJM model, 566 and binomial options pricing models, 375 with caplets, 579 with European puts, 352–53 and HJM libor model, 623 illegal, 124–26 Arbitrage profits in binomial options pricing model, 372–73 for commodities with storage costs/convenience yields, 244 for foreign currency forwards, 242 and forward prices, 220 for forwards with fixed dollar dividends, 233 and law of one price, 212–13 and put–call parity, 341 for synthetic indexes, 238 Arbitrage tables for commodities with storage costs/convenience yields, 245 for cost-of-carry model, 220–21 for currency forward price, 243 for forward price of dividend-paying stock, 235

for forwards with market imperfections, 250 and options price restrictions, 346, 349–52, 355 and put–call parity, 340 for zero-value portfolios, 218 Arbitrageurs, 57 Arbitration, 56 Ask price and banker’s discount yield, 46–47 of stock, 56 of T-bills, 45–47 Asset allocation, portfolio risk, 14 Asset-backed securities (ABSs), 38, 498 Asset-backed securities credit default swaps (ABS CDSs), 148 Assets, 14 cash flows and values of, 131–32 consumption, 214 with continuously compounded dividend yield, 64 current, 658 financial, 7 intrinsic value of, 215, 375, 424 investment, 214 long-term, 658 production, 214 real, 7 Assignment, in swap closing, 544 Associated persons (APs), 173 Asymmetric information, 677 At-the-money (term), 95, 101 Auctions, Treasuries, 38–39 Australian Options Market, 295 Automated trading, 58, 295 Average forward rate volatility, 632–36 Backward induction, 396–97, 601 Backwardation, 246–49 Balance sheet, 15, 16, 658, 659, 670 Banging the close, 204–5 Bankers, financial engineering by, 135 Banker’s discount yield, 46, 47 Bankers Trust, 539, 540 Bankhaus Herstatt, 669 Barings Bank, 156, 260n1

720

SUBJECT INDEX

Barrier options, 449 Basel Committee on Banking Supervision, 12–13, 17, 668–78 Basis, 185, 187, 264 Basis point (bp), 37 Basis risk, 264–67 BBA, see British Bankers’ Association libor vs. libor rate index, 47–48 Bear raids, 124 Bear spreads, 321–23 Bearish trades, 65, 93 BEP (breakeven point), 316 Bid/ask (bid/offer) spread, 56, 251 Bid price, 18, 46, 56 Binomial (term), 369, 687 Binomial coefficients, 403, 688 Binomial Heath–Jarrow–Morton (binomial HJM) model, 558–89, 592–615 assumptions with, 565–70, 593–96 caplet pricing with, 620–22 equal pseudo-probability condition for, 575–76, 586–87 for forwards, 607–8 for futures, 608–10 and history of interest rate derivative models, 564–65 in history of options pricing, 369–71 and HJM libor model, 613–15 and interest rate derivatives, 560–64 multiperiod, 607 single-period, 570–85 for swaptions, 611–12 Treasuries in, 37 two-period, 597–606 Binomial interest rate derivatives pricing model, 384 Binomial option pricing (BOP) data, 394–95 Binomial options pricing model, 367–88, 392–418 for American options, 410–12 applications and uses of, 368–69 assumptions with, 374–76 BOP data in, 394–95 example, 372–74 and history of options pricing, 369–71 multiperiod, 403–9 and no-arbitrage argument, 384–88 for options with dividends, 409–10 single-period, 376–83 spreadsheet programs for implementing, 413–17

stock price path in, 393–94 and stock price tree, 395–96 two-period, 396–402 Binomial process assumption, 376, 567, 593 Binomial stock price evaluation, 393–94 Binomial theorem, 687–88 Binomial trees arbitrage-free conditions for, 573, 597 in Microsoft Excel, 413 in options pricing model, 377–78 Black Monday, 316 Black’s model of caplet pricing, 620, 621 Black–Scholes–Merton (BSM) model, 8, 12, 94, 422–59, 462–87, 621 actual vs. pseudo-probabilities for, 433–34 for American options, 447–48 assumptions with, 424–28 and binomial options pricing model, 369, 407–8 calibration of, 474–80 continuous compounding in, 34 derivations of, 454–56, 486–87 with dividends, 444–45 for European options, 486–87 for exotic options, 448–49 for foreign currency options, 446–47 formula, 429–32 Greeks in, 434–40 hedging limitation of, 480–81 and hedging of Greeks, 463–69, 482–87 and hedging portfolios, 469–70 in history of options pricing/trading, 291, 370–71 HJM libor model vs., 630–31 inputs for, 440–44 and lognormal assumption for stock prices, 451–59 martingales in, 432–33 and pricing/hedging argument, 428–29 risk-neutral valuation in, 433 and vega hedging, 470–73, 485–86 Block trades, 60 Bond-equivalent yield, T-bill, 45–47 Bonds; see also Zero-coupon bonds (zeroes) callable, 131 cheapest-to-deliver, 530 convertible, 130 coupon, 43, 519–20 defined, 7

721

Eurobonds, 8 extendible, 130 inflation-indexed, 43 modified duration for, 508, 509 Treasury, 43–45, 530 Boomsters, 168–69 BOP (binomial option pricing) data, 394–95 Borrowing, role of derivatives in, 5 Borrowing rates, 251 Bottom straddle, 330 BOVESPA, 295 Bp (basis point), 37 Breakeven point (BEP), 313 Bretton Woods system, 6 Broker loan rate, 67 Brokerage commissions, 249, 251 Brokers in futures market, 173–74 introducing, 173 in stock market, 56–57 of swaps, 135 Broker’s call, 67 BSM model, see Black–Scholes–Merton model Budget deficit, 38 Build and break approach, 130–31 Bull spread, 321–22, 324, 326 Bullish trades, 65, 93 Butterfly spread, 325, 326 Buyers options, 93, 95–98 swap, 545 Buying (long) hedges, 89, 266 Buying the spread, 187 Buyout, swap, 544 Buy–write options, 317, 318 BXM (CBOE S&P 500 BuyWrite Index), 317 Calendar spread, see Time spread Calibration, 444, 474–80 defined, 636 and hedging errors, 476–77 and implied volatilities, 477–80 and theoretical vs. econometric models, 474–75 using, 475–77 Call backspread, 327n5 Call loan rate, 67 Call options (calls), 95–100; see also Put–call parity American, 99–100, 344–46, 360–62 arbitrage opportunities with, 350

722

SUBJECT INDEX

binomial pricing model for, 372–74, 400, 401 BSM model for, 431 on commodity futures, 296 covered, 313, 317–20 on currency, 446–47 defined, 93 delta and gamma for, 437, 438 delta hedging errors with, 465, 468–69 down-and-out, 449 early exercise of, 360–62 European, 347, 352, 372–74, 400, 415–16, 431, 438, 446, 447 exchange-traded, 105–6 on indexes, 445–46 on interest rates, see Caplets intrinsic and time values of, 98–99 long, 298, 315–16, 318, 319, 344–45, 467 payoffs and profit diagrams for, 95–98 price bounds for, 99–100, 347–49 pricing, with Microsoft Excel, 414–16 risk-neutral pricing for, 401 short, 314–18, 344–45 and strike price, 350, 352 synthetic, 372, 397–400 Call ratio spread, 327n5 Call spreads bearish, 323 bull, 321–22, 324, 326 one-by-two, 325 and reinsurance, 327 Callable bonds, 131 Callable swaps, 550 Cancel orders, 108 Cancel replace orders, 108 Cancellable swaps, 550 Capital asset pricing model, 8 Capital budgeting, 506, 659 Capital gains, 7, 24 Capital losses, 7 Caplet and floorlet pricing (CFP) data, 627 Caplet pricing, 619–20 with Black’s model, 621–22 history of, 620–22 with HJM libor model, 627–31 with single-period binomial HJM model, 576–80 with two-period binomial HJM model, 601–4 Caplet–floorlet parity, 639–41 and floorlet pricing, 583, 605–6 proof of, 588–89

Caplets, 559–61, 605 delta hedging of, 643–45 delta of, 642 gamma hedging of, 646–47 gamma of, 642 hedge ratio for, 581 maturity of, 577 payoffs for, 562 risk-neutral valuation of, 581, 602–4, 637–38 Caps, 559, 560, 638–41 hedging interest rate risk with, 562–63 HJM libor model for, 638–41 payoffs for, 563 speculation with, 627–28 on swaps, 549 Carrying charges, 210 Cartels, 196 Cash-and-carry argument, 210 Cash and margin accounts, see Margin accounts Cash dividends, 62 Cash flows and asset values, 131–32 compounding and discounting of, 31–37 in currency swaps, 139, 142–43 for dividend-paying stock, 234 of FRAs, 522 in interest rate swaps, 136–38 intermediate, 215, 375, 424, 566, 623 in margin accounts, 179–80 present value of, 35 Cash prices, see Spot prices Cash-secured puts, 316 Cash-settled futures contracts, 160 Cash settlement, 74, 175 Cause-and-effect relationships, in theoretical models, 474 CBOE, see Chicago Board Options Exchange CBOE S&P 500 BuyWrite Index (BXM), 317 CBOE Volatility index (VIX), 294, 480 CBOT, see Chicago Board of Trade CBT, see Chicago Board of Trade CDS spread, 148 CEA (Commodity Exchange Act), 197 Central limit theorem (CLT), 453–54 CFMA (Commodity Futures Modernization Act of 2000), 198–99 CFP (caplet and floorlet pricing) data, 627

CFTC, see Commodity Futures Trading Commission Chase Manhattan Bank, 540 Cheapest-to-deliver bonds, 530 Chicago Board of Trade (CBOT; CBT) futures regulation at, 196–97 goal of, 12, 259 history of, 157–59 interest rate derivatives of, 6, 520–21 trading at, 83 Chicago Board Options Exchange (CBOE), 105, 290 call option example, 105–6 opening of, 293 trading on, 296 and VIX, 294, 480 Chicago Mercantile Exchange (CME; the Merc) futures trading at, 83 history of, 158 IMM division, 6, 159 interest rate derivatives on, 521 China, futures trading in, 86 China Financial Futures Exchange, 86 Clearance fees, 166 Clearing (of trades) for call options, 105–6 for futures, 84 for securities, 59, 61 Clearing members, 59 Clearinghouses, 83, 84, 158, 166–67 Closed-end companies, 120 Closed-end funds, 119–20 Closed-form solutions, 94n1, 641 Closing (of trades), 205 for call options, 105–6 early, 181 for futures, 85, 181 for swaps, 544 Closing call, 166 Closing price (closing range), 166 Closing transactions, 174 CLT (central limit theorem), 453–54 CME, see Chicago Mercantile Exchange CME ClearPort, 162 CME Group, 195, 197 CMSs (constant maturity swaps), 550 Code of SWAPS, 542 Coherent risk measures, 667 Collars, option, 323–34, 564 Combination strategies, options trading, 329–31 COMEX, see Commodity Exchange Commercial traders, futures, 205

SUBJECT INDEX

Committee on Uniform Security Identification Procedures (CUSIP) number, 44 Commodities, 161n1 Black’s model for pricing options on, 621–22 hedging for spot commodity positions, 269–70 intercommodity spreads, 185, 188 intracommodity spreads, 185, 187, 188 with storage costs and convenience yields, 245 in supply chains, 86 Commodity Exchange (COMEX), 161, 162, 164 Commodity Exchange Act (CEA), 197 Commodity Exchange Commission, 197 Commodity funds, 174 Commodity futures, 161, 296 Commodity Futures Modernization Act of 2000 (CFMA), 198–99 Commodity Futures Trading Commission (CFTC), 81, 197 enforcement actions by, 201 listing requirements of, 55 registration with, 173 regulation by, 198–200 on short squeezes, 126 Commodity Futures Trading Commission Act, 160, 197 Commodity index traders, 205 Commodity-indexed notes, 134 Commodity markets, 205–6 Commodity pool operators (CPOs), 174 Commodity pools, 174 Commodity price indexes, 167–69 Commodity price risk, 15 Commodity swaps, 146–47 Commodity trading advisors, 174 Common options price (COP) data, 313–14 Competitive markets, 193 and binomial HJM model, 566 and binomial options pricing models, 375 and BSM model, 424 and cost-of-carry model, 215 and HJM libor model, 623 Complete markets, 156, 290, 399, 425 Compound interest rates, 27–30 Compounding cash flow, 31–37 continuous, 687 Condor spreads, 325, 326

Confirmations, of Master Agreements, 542 Congestion, 126 Consols, 143 Constant maturity forward rates, 633, 634 Constant maturity swaps (CMSs), 550 Constant parameters, hedging, 470–72 Consumer price index (CPI), 23 Consumption assets, cost-of-carry model for, 214 Contango, 246–49 Contingent claims models, 669–71 Contingent orders, 107, 108 Continuous compounding, 687 Continuous trading assumption, 426, 427, 626 Continuously compounded dividend yield, 64 Continuously compounded forward rates, 625, 626 Continuously compounded interest rates, 27, 29–34, 37, 218 Continuously compounded return, 30, 451, 452 Contract months, for Eurodollar futures, 527 Contract size, 162 Convenience yield (convenience value), 244–45 Convergence of basis, 185 Conversions, 292 Convertible bonds, 130–31 Convexity hedging, 512 COP (common options price) data, 313–14 Cornering the market, 125–26, 197 Corporate financial managers, 497 Corporate financial risk management, 14–17, 658–60 Corporate hedging, 259–64 Correlations, in econometric models, 474 Cost-of-carry model, 209–25, 229–51 arbitrage table approach, 220–21 assumptions with, 214–16, 230–32 with backwardation/contango, 246–49 and dividend-paying stocks, 232–43 with dividend yield, 235–36 with dollar dividends, 232–35 and forward maturities, 224–25 for forwards at intermediate dates, 221–24

723

law of one price in, 210–14, 217 and market imperfections, 249–51 and nothing comes from nothing principle, 213–14 setup of, 216 with storage costs and convenience yields, 243–46 Cost of construction, options, 372–73, 380 Cost of exercising (term), 436 Costs-of-carry, 210 Counterparties, 60 Counterparty risk, see Credit risk Coupon bonds, 43, 503, 519–20; see also Zero-coupon bonds (zeroes) Coupons, 7, 43 Covered calls, 313, 317 Covered interest rate parity, 241 Covered puts, 319–20 Covered trading strategies, for options, 314, 317–20 Cox–Ross–Rubinstein binomial model, 407 CPI (consumer price index), 23 CPOs (commodity pool operators), 174 Credit default swaps (CDSs), 147–48, 676 Credit events, credit default swaps and, 147–48 Credit risk, 13, 272–73, 660 and binomial HJM model, 566 and binomial options pricing models, 375 and BSM model, 424 and cost-of-carry model, 214 hedging, 148 and HJM libor model, 622 options pricing with, 371 reduced-form models of, 673–76 structural models of, 669–70 Credit spread, 321 Credit support, for swaps, 542 Critical assumptions, of models, 679 Cross-currency settlement risk, 669 Cross-currency swaps, see Currency swaps Cross-hedges, 264, 269–70 Cross-rates, currency, 138 Crude oil, 204 Cum-dividend (term), 62, 63 Cumulative standard normal distribution, 688–90 Currency forwards, 78–79

724

SUBJECT INDEX

Currency risk, for businesses, 15 Currency swaps fixed-for-fixed, 139–45 plain vanilla, 94–95, 139–40, 145 Current assets, 658 CUSIP number, 44 Daily price limit (maximum daily price fluctuation), 162 Daily return variance, 663 Daily settlement, futures, 84, 176–80 Dalian Commodity Exchange, 86 Dark pools of liquidity, 61–62 DARPA (Defense Advanced Research Projects Agency), 193 Day orders, 108 Day traders, 57 DCMs, see Designated contract markets De Beers Group, 196 De Moivre-Laplace theorem, 455 Dealers futures, 173–74 market maker, 56 stock, 56–57 swap, 205, 541 Dealing rooms, OTC market, 79 Debit spread, 321 Decay, options, 313 Default, 259 Default-free coupon bonds, yield for, 503 Default risk, see Credit risk Defense Advanced Research Projects Agency (DARPA), 193 Deferred swaps, 550 Deliverable supply, 125 Delivery (maturity) date of forwards, 73, 78 of futures, 162 and interest rate caplets, 627–28 of options, 296–97, 395–96, 413 Delivery period, futures, 162 Delivery price, forward, see Forward price Delta (hedge ratio), 428, 643 and BSM model, 463 of caplets, 579, 581, 643–44 interpreting, 478 of options, 435–39 in options pricing, 379–80 463, 482 of portfolios, 469 sensitivity of, see Gamma Delta- and gamma-neutral portfolios, 484–85

Delta- and vega-neutral portfolios, 485–86 Delta hedging, 463–66 and constant parameters, 470–72 with HJM libor model, 641–47 of long calls, 467 of portfolios, 469 Delta-neutral portfolios, 483, 643–44 Depreciation, US dollar, 138 Derivative transaction execution facilities, 198 Derivatives (derivative securities), 3–12 acceptance of pricing models for, 475–76 application and uses of, 10–11 build and break approach for, 130–31 financial innovation with, 5–7 future of, 679–80 investors’ views of, 9–10 pricing of, 12, 118 terminology for, 7, 9 valuation of, 12 Derivatives market expansion of, 5–6 global, 4 history of, 3–5 Designated contract markets (DCMs), 198 Diagonal spreads, 321 Diamond cartel, 195–96 Digital options, 448–49 Direct terms, for currency quotes, 138 Discounting, cash flow, 31–37 Discrete trading, 427 Diversifiable risk, 14 Diversification, value-at-risk and, 665–66 Dividend date, 62, 410, 445 Dividend rates, 410, 445 Dividend yield, 63–64 cost-of carry model with, 235–36 forward price of index with, 237–39 put–call parity for options with, 344 Dividends, 62–64 and binomial options pricing model, 409 and BSM model, 444–46 cash, 62 defined, 7 dollar, 232–35, 344, 410, 444–45 and early exercise, 360–62 forwards on stocks with, 232–43 options adjustments for, 299–300 options with, 409–10, 444–46

and put–call parity, 344 synthetic indexes with, 236–39 Documentation, swap, 543–44 Dodd–Frank Wall Street Reform and Consumer Protection Act, 199 Dojima Rice Exchange, 155, 157 Dollar, US, 6, 138 Dollar dividends and binomial options pricing model, 410 and BSM model, 444–45 and cost-of carry model, 232–35 and put–call parity, 344 Dollar return, 31–37, 378 Doomsters, 168–69 Double privilege, 329 Dow Jones Industrial Average (DJIA), 121 Dow Jones–AIG Commodity Index Family, 167 Down-and-out call options, 449 Down factors, 377, 378, 408 Duration, 507–10 in active management, 512 and modified duration hedging, 510–12 and price change in bonds, 508–10 Duration-neutral portfolios, 512 Dynamic hedging, 272 Dynamic market completion, 399 EAR (effective annual interest rate), 30n3 Early exercise, of American options, 357–62 ECN (electronic communications network), 59, 62 Econometric models, 474–75 Econometrics, 281 Economic terms, of swap transactions, 542 Effective annual interest rate (EAR), 30n3 Efficient market hypothesis (EMH), 118–19 Efficient markets, 118–19 EFP (exchange for physicals), 175 Electronic communications network (ECN), 59, 62 ELNs (equity-linked notes), 134 EMH (efficient market hypothesis), 118–19 Employee stock options, calls against, 318–19

SUBJECT INDEX

Equal pseudo-probability condition, 572 for binomial HJM model, 575–76, 597 proof of, 586–87 Equilibrium models, 387 Equity shareholder’s, 15, 658 value of, 659–60, 670 Equity carve-outs, 343 Equity-linked notes (ELNs), 134 Equity options trading, 289; see also Options trading Equity swaps, 146 Estimated trading volume, on price quotes, 166 ETFs (exchange-traded funds), 168, 235–36 ETNs (exchange-traded notes), 168 Euribor (Euro Interbank Offered Rate), 603–4 Eurobonds, 8 Eurodollar futures, 160, 524–30, 560, 561, 608–10 Eurodollar futures rate, 610–11 Eurodollar rate, 48, 499 Eurodollars, 8, 499 development of, 8 in London interbank market, 47–48 in synthetic interest rate swaps, 550–52 Treasury–Eurodollar spread, 48, 500, 554 yield and swap curve for, 554 Europe, medieval, 156 European options; see also specific pricing models, e.g.: Black–Scholes–Merton (BSM) model American vs., 347 arbitrage opportunities with, 352–53 on currency, 447 delta and gamma for, 435, 438 exercise of, 94 on indexes, 445–46 price bounds for, 347, 348, 352 profit diagrams for, 312–13 put–call parity for, 338–42, 344 strike price of, 352 European Options Exchange, 295 European terms, for currency quotes, 138 Even lot, 162 Ex-dividend (term), 62, 234 Excess margin, 179 Exchange for physicals (EFP), 175 Exchange rate, 138

Exchange-traded funds (ETFs), 168, 235–36 Exchange-traded notes (ETNs), 168 Exchange-traded options, 105–6, 289–90 Exchanges; see also specific exchanges, e.g.: New York Stock Exchange (NYSE) futures, 81, 83–88, 161, 166–67, 201 options, 295–96 securities, 55–57 stock, 60–61 Exclusions, insurance policy, 327 Execution (of trades) for futures, 83–85 for securities, 58 for stock, 60 Exempt boards of trade, 198 Exercise by exception (ex-by-ex) procedure, 301 Exercise prices, see Strike prices Exercise value, of options, 98, 103 Exercising, of options, 93, 300 Exotic derivatives, 94 Exotic options, 448–49 Exotic swaps, 94 Expectations hypothesis, 502, 504–6 Expected return, value-at-risk and, 664 Expiration date, option, 93, 95, 348, 352 Expiration month, option, 296 Expiration process, option, 301 Exponents, 685, 686 Extendible bonds, 130 Face value, of coupon bonds, 43 Factorials, 686 Factors (term), 584 Failure to deliver, 41 FASB 133 (Accounting for Derivative Instruments and Hedging Activities), 16 FCM (futures commission merchant), 84, 173 February cycle, 297 Fed, see Federal Reserve System Federal debt, 38–39 Federal funds market, 48 Federal Reserve System, 6, 67 Fees, option, 93 Final settlement, of Eurodollar futures, 527 Finance, 8 Financial assets, 7 Financial crisis, see Great Recession (2007-2009)

725

Financial distress costs, 259 Financial engineering, 18, 131–34; see also specific instruments, e.g.: Swaps Financial futures (financials), 155–56, 161 Financial innovation, 5–7 Financial institutions interest rate derivatives at, 497 risk management from perspective of, 17 Financial managers, corporate, 497 Financial risk, 4 Firms, hedging by individuals vs., 257–58 First delivery day, 162 First notice day, futures, 174 Fixed-for-fixed currency swaps, 139–45 Fixed-income securities, 23; see also Bonds Fixed-rate loans, 136–38, 545–47 Fixed-rate mortgage (FRM), 498 FLEX options, 294 Floating-rate loans, 136–38, 545–47 Floor trading, illegal arbitrage in, 124–26 Floorlets, 559–61; see also Caplet–floorlet parity binomial HJM model for, 583, 605–6 caplet and floorlet pricing data, 627 HJM libor model for, 639 maturity of, 577 payoffs for, 562 Floors defined, 559, 560, 638 HJM libor model for, 638–41 payoffs for, 562 Foreign currency forwards, 239–43 Foreign currency options, 446–47 Forex (foreign exchange) markets, 78–80, 138 Forex (FX) swaps, 138–39 Forward contracts, see Forwards Forward exchange rates, 240–43 Forward market model, 622; see also Heath–Jarrow–Morton (HJM) libor model Forward price (F), 73, 76; see also Cost-of-carry model and arbitrage profit, 219–20 defined, 11 future vs., 180–85 and maturity, 183–85, 224–25 and spot price, 246–49 Forward rate agreements (FRAs), 521–23, 560, 561

726

SUBJECT INDEX

and forward rates for interest rate derivatives, 530–31 rates for Eurodollar futures vs., 610–11 in synthetic interest rate swaps, 550–52 two-period binomial HJM model for, 607 Forward rate agreements rate Eurodollar futures rate vs., 610–11 for HJM libor model, 631–32 Forward rate curve, 516 Forward rates, 513–20 average volatility of, 632–36 computing, 613–14, 623–25 constant maturity, 633, 634 continuously compounded, 625 defined, 514 for discrete time intervals, 623 for future period, 566 futures vs., 610 in HJM libor model, 623, 637 for interest rate derivatives, 530–31 interpretations of, 516–17 limit distribution of, 614 and martingales, 637 from swap rates, 553 Forward swaps, 550 Forwards (forward contracts), 73–78; see also Forward rate agreements (FRAs); Interest rate swaps application and uses of, 155–56 binomial HJM model for, 607–8 defined, 11 on dividend-paying stocks, 232–43 futures vs., 81–83, 180–85, 272–73 hedging with, 88–89, 272–73 at intermediate dates, 221–24 maturities of, 224–25 terminology for, 73–78 timeline for, 75 Forwards market history of, 156–57 over-the-counter, 78–80 Fractional stock splits, 299–300 Free-market environmentalism, 168 Frictions, see Market imperfections FRM (fixed-rate mortgage), 498 Front running, 124 Fundamental analysis, 119 Fundamental value and cost-of-carry model, 215 and price bubbles, 375, 424 Futures (futures contracts), 81–89 application and uses of, 155–56 binomial HJM model for, 608–10

cash-settled, 160 Eurodollar, 160, 524–29, 561, 563, 608–10 features of, 161–64 forwards vs., 81–83, 180–85, 272–73 hedging with, 88–89, 194, 264–68, 272–73 HJM libor model for, 647 interest rate, 524–30 price quotes for, 164–66 types of, 159–61 Futures accounts, 176 Futures commission merchant (FCM), 84, 173 Futures contracts, see Futures Futures market, 192–206 brokers and dealers in, 173–74 and commodity markets, 205–6 and commodity price indexes, 167–69 exchanges and clearinghouses in, 166–67 history of, 156–61 key conditions of, 193–94 manipulation in, 202–5 regulation of US, 196–201 speculation in, 195–96 Futures options, 160, 295–96 Futures prices, 524, 525 arbitrage-free, 608–9 forwards vs., 180–85 and maturity, 183–85 price discovery, 194 Futures rates arbitrage-free, 609–10 forward vs., 610–11 Futures trading, 172–89 on exchanges, 83–88 margin accounts and daily settlement for, 176–80 spreads in, 185–88 trading futures, 174–76 FX (forex) swaps, 138–39 Gambling, with derivatives, 5 Gamma of call options, 437–39 of caplets, 642 of European options, 435–39 interpreting, 434–35 of portfolios, 469–70 of put options, 437–39 Gamma hedging, 466–69, 483–85, 641–42, 646–47 Gasoline futures, manipulation of, 204

Gearing, see Leverage Geometric Brownian motion assumption, 426 Gilt securities, 143 GNMA (Government National Mortgage Association), 160, 520 Gold futures, 162–64 basis risk with, 265–66 daily settlement of, 177–78 price quotes for, 164–66 trade records, 88 Gold market bonds indexed to prices in, 132–33 manipulation in, 202 Goldman Sachs and Company, 312 Goldman Sachs Group, 656 Good-until-canceled (GTC) orders, 108 Government interest rate derivatives for, 497 Government National Mortgage Association (GNMA), 160, 520 Grade specifications, in futures trading, 163 Grain Futures Act, 197, 198 Grain Futures Administration, 197, 198 Grantors, see Writers, option Great Moderation, 6 Great Recession (2007–2009) credit default swaps in, 540 derivatives in, 3, 5, 6 model misuse in, 463 scenario analysis by banks in, 666 Greeks in Black–Scholes–Merton model, 434–39 hedging, 463–69, 482–85, 642–47 Gross domestic product, 17 GTC (good-until-canceled) orders, 108 Heath–Jarrow–Morton (HJM) libor model, 12, 618–49 and actual vs. pseudo-probabilities, 638 for American options, 647 assumptions with, 622–27 and binomial HJM model, 613–15 Black–Scholes–Merton model vs., 630 and caplets, 619–22, 627–31 for caps and floors, 638–41 delta hedging in, 643–45 forward rates and martingales in, 637 for futures, 647 hedging Greeks with, 640–47 inputs for, 631–36

SUBJECT INDEX

risk-neutral valuation in, 637–38 for swaptions, 647 Heath–Jarrow–Morton (HJM) model, see Binomial Heath–Jarrow–Morton (binomial HJM) model Heating oil futures, manipulation of, 204–5 Hedge ratio, see Delta Hedgers, 57 Hedging, 256–81; see also Delta hedging of airline fuel costs, 146, 147 and binomial options pricing model, 399 and BSM model, 480–81 calibration and errors in, 476–77 with caps, 562–63 computing optimal hedge ratio, 273–75 of constant parameters, 470–72 convexity, 512 corporate, 259–64 with currency forwards, 79 of default risk, 148 defined, 3 with derivatives, 5, 10–11 duration, 512–13 dynamic, 272 with Eurodollar futures, 526 by firms vs. individuals, 257–58 with forwards, 88–89, 272–73 with futures, 88–89, 272–73, 264–68, 272–73 gamma, 466–69, 483–85, 646–47 of hurricane risk, 327–28 of interest rate risk, 641 modified duration, 510–12 in options trading, 317–21 of portfolios, 469–70 pricing and hedging argument, 428–29 rho, 439 risk-minimization, 268–72 of risks, 16 and synthetic calls, 372 vega, 439, 470–73, 480, 485–86, 641 Herstatt risk, 669 High-frequency trading, 425; see also Algorithmic trading (algo) High price, futures, 166 Historical data, pricing models and, 476 Historical volatility, 441–44, 633–36 HJM models, see Binomial Heath–Jarrow–Morton (binomial HJM) model;

Heath–Jarrow–Morton (HJM) libor model Horizontal spreads, 185, 321 Housing, residential, see Residential housing market Hunt Brothers, market manipulation of, 202–3 Hurricane risk, hedging, 327–28 Hybrid debt, 134 IB (introducing broker), 173 IBM (International Business Machines), 297, 299–304, 539–40 Identical distribution, of stock returns, 452 iid property, of stock returns, 452 Illegal arbitrage opportunities, 124–26 IMM (International Monetary Market) division, 6, 159 Imperfect hedges, 264, 269–71 Implied volatility, 441 average forward rate, 636 and calibration, 477–80 computing, 478–79 historical vs., 632 In-the-money (term), 95, 101 Independence, of stock returns, 452 Index arbitrage, 121–23 Index futures, 160 Index mutual fund, 235 Indexes; see also Stock indexes European options on, 445–46 for interest rate derivatives, 560 market-cap weighted, 123 synthetic, 236–39 total return, 236, 238 India futures trading in, 86–87 Indirect terms, for currency quotes, 138 Individuals hedging by firms vs., 257–58 interest rate derivatives for, 497 Induction, backward, 396–97, 601 Inflation, 23 Informationally efficient markets, 118 Initial margin, 67, 163, 176 Initial public offering (ipo), 54 Input price risk, 88–89 Insiders, 119 Insolvent firms, 659 Institutional investors, in Treasuries, 40 Institutional Networks (Instinet), 62 Insurance, 5, 290 Integer multiple stock splits, 299

727

Interbank market, 55, 78; see also Over-the-counter (OTC) markets Intercommodity spread, futures, 185, 188 Interest, in cost-of-carry model, 218–20 Interest rate collars, 549 Interest rate derivatives (IRDs), 496–534; see also Interest rate swaps about, 497–99 and binomial HJM model, 560–64 forward rate agreements, 521–23 forward rates for, 513–20, 530–31 FRA rates for, 530–31 history of, 520–21, 564–65 interest rate futures, 524–30 interest rate risk management with, 507–13 and Treasuries, 43 and yields, 499–507 Interest rate futures, 524–30 Interest rate parity, covered, 241 Interest rate risk, 15, 562, 641 Interest rate risk management, 507–13 Interest rate swaps, 136–38, 538–55 and forward rate agreements, 550–54 history of, 539–42 institutional features of, 542–44 plain vanilla, 94, 136–38 valuation of, 544–49 variations of, 549–50 and yield–swap curve relation, 552, 554 Interest rates, 22–50 about, 23–24 compound, 25–31 continuously compounded, 27–30, 36–37, 219 discounting and compounding cash flows with, 31–37 libor vs. libor rate index, 47–48 and measuring time, 26 and option value, see Rho and rate of return, 24–25 and rounding/reporting of numbers, 25 simple, 27, 27–31 term structure of, 371, 502, 567–68, 594, 647 and Treasuries, 37–47 uncertainty about, 375–76, 425 Intermediaries, forward transaction, 80 Intermediate cash flows, 215, 375, 424, 566, 623 Intermediate dates, forwards at, 221–24

728

SUBJECT INDEX

Internal rate of return (IRR), see Yield Internalization, of orders, 59 International Business Machines, see IBM International Commercial Exchange, 159 International Monetary Market (IMM) division, 6, 159 International Organization for Standardization (ISO), 79n3 International Organization of Securities Commissions (IOSCO), 12, 13 International Securities Exchange, 295 International Swap Dealers Association, 138, 541 International Swaps and Derivatives Association (ISDA), 138, 541 Intracommodity spread, futures, 185, 187, 188 Intrinsic value and cost-of-carry model, 215 of options, 98–99, 103 and price bubbles, 375, 424–25 Introducing broker (IB), 173 Investment assets, cost-of-carry model for, 214 Investment companies, portfolio risk management by, 14 IOSCO (International Organization of Securities Commissions), 12, 13 IPO (initial public offering), 54 IRDs, see Interest rate derivatives IRR (internal rate of return), see Yield ISDA, see International Swaps and Derivatives Association ISDA Agreement Structure, 542 ISO (International Organization for Standardization), 79n3 Issuers, see Writers, option Ito’s lemma, 487 ITT Corporation, 540 January cycle, 297 Jarrow–Rudd binomial model, 407 Jensen’s inequality, 610 J. P. Morgan, 540–41, 662 J. P. Morgan Chase, 498 Kansas City Board of Trade, 160 Last delivery day, 162 Last notice day, 174 Last trade price, 165 Last trading day, futures, 162, 174, 527

Last updated, on price quotes, 166 Law of large numbers, 327 Law of one price, 114, 117 cost-of-carry model from, 207–11, 216–21 and no-arbitrage principle, 117 LE (local expectations) hypothesis, 504, 600 LEAPS (Long-Term Equity AnticiPation Securities), 294 Least squares method, 281 Legal risk, 13, 678 Lending, role of derivatives in, 5 Lending rates, 251 Leverage with derivatives, 5, 9 margin for, 66 and put–call parity, 342 Liabilities, 14, 658 Libid (London Interbank Bid Rate), 48 Libor (London Interbank Offered Rate), 47–48, 50, 499 Libor market model, 622; see also Heath–Jarrow–Morton (HJM) libor model Libor vs. libor rate index, 47–48 Lifetime highs, futures, 166 Lifetime lows, futures, 166 Limit buy orders, 108 Limit distribution, forward rate, 614 Limit order book, 62 Limit orders, 62, 105, 108 Linear regression models, 281 Liquidation, payoffs at, 132, 133 Liquidation-only trading, 203 Liquidity dark pools of, 61–62 in securities market, 57 Liquidity preference hypothesis, 504 Liquidity risk, 13, 660, 668, 677–78 Listing requirements, stock exchange, 55 Loans fixed-rate, 136–38, 545–46 floating-rate, 136–38, 545–46 interest rate swaps with, 136–38 secured, 67 Local expectations (LE) hypothesis, 504–5, 600 Locked-in transactions, 60 log(x), 685 Logarithmic (continuously compounded) return, 30, 451–52 Logarithms, 685–86

Lognormal assumption (for stock prices), 426, 451–59 Lognormal distribution, for simple forward rate, 626 London Interbank Bid Rate (libid), 48 London Interbank Offered Rate (libor), 47–48, 50, 499 London Stock Exchange, 59, 157 London Stock Exchange Committee, 291 London Traded Options Market, 295 Long calls in covered trading strategies, 318, 320 delta and gamma hedging of, 467 margin requirements for, 298 profit diagrams for, 315–16 put–call parity for, 344–46 Long hedges, 88, 266 Long positions (longs) in covered trading strategies, 317–20 in derivatives, 74 in 1:2 hedges, 320 on options, 94 in options market, 107 payoffs for forwards with, 77–78 and put–call parity, 339 in stock, 314 Long puts in covered trading strategies, 319 margin requirements for, 298 profit diagrams for, 315, 316 put–call parity for, 339, 345 Long straddles, 330–31 Long strangles, 330–31 Long-term assets, 658 Long-Term Capital Management (LTCM), 677 Long-Term Equity AnticiPation Securities (LEAPS), 294 Loss distribution, 660–62 Low price, futures, 166 LTCM (Long-Term Capital Management), 677 Maintenance margin, 67, 163, 178 Malthusian doctrine, 168 Managing books (term), 18, 541 Margin for futures, 163 in history of futures, 158 for options trading, 298–99 for portfolio of futures, 182 Margin accounts, 66–68, 176–80 Margin calls, 67

SUBJECT INDEX

Mark-to-model accounting, 221 Marked-to-market, futures, 176, 272 Market buy orders, 107 Market-cap weighted index, 123 Market capitalization, 122 Market close (closing call), 166 Market corners, 125–26, 202 Market imperfections (market frictions) arbitrage opportunities from, 115 and binomial HJM model, 565 and binomial options pricing models, 374 and cost-of-carry model, 214, 247, 250 defined, 95 and futures/spot prices, 183 and HJM libor model, 622 and put–call parity, 342–46 Market maker dealers, 56 Market manipulation futures, 202–5 and option pricing, 439–40 options, 305–6 Market model, 565, 622; see also Heath–Jarrow–Morton (HJM) libor model Market orders, 60, 105, 107 Market price, calibration and, 476 Market risk, 13, 660, 668 Market segmentation hypothesis, 504 Marking the close, 204 Marking-to-market (term), 84 Married puts, 318, 319 Martingale pricing and binomial options pricing model, 369 and BSM model, 456 defined, 379 Martingales in Black–Scholes–Merton model, 432–33 and forward rates, 637 in HJM libor model, 637 and single-period binomial HJM model, 573–75 in single-period binomial options pricing model, 378–79 and two-period binomial HJM model, 599–600 and zero-coupon bond prices, 573–75, 599–600 Master Agreement, ISDA, 542 Maturity of caplets and floorlets, 577

and futures prices, 183–85 and option prices, 355 prices at, 395–96, 415 and prices of forwards, 224–25 spot price at, 183 of swaps, 135 Maturity date, see Delivery date Maximum daily price fluctuation, 162 MCX (Multi Commodity Exchange), 87 Mean, of constant maturity forward rates, 634 Mean-variance approach, 268–71 Merc, see Chicago Mercantile Exchange Microsoft Excel; see also Spreadsheet software computing optimal hedge ratio in, 273–75 multiperiod binomial options pricing models in, 413–17 option pricing project, 491–93 Miller and Modigliani (M&M) propositions, 8, 117, 257 Minimum price fluctuation, 162, 190, 527 Minimum-variance hedge ratio (h), 276–81 Mizuho Securities Company, 678 Model risk, 679–80 Models definition of, 214 Modified duration, 507–10 Modified-duration hedge ratio, 511 Modified duration hedging, 510–12 Money market account (mma), 372 dollar return from, 31 moving cash flows across time with, 34–35 and spot rate, 565, 595 Money market yield basis, 549 Monotonicity, of risk measures, 667 Montreal Stock Exchange, 295 Mortgage-backed securities, 498 Mortgages, 498 Multi Commodity Exchange (MCX), 87 Multifactor models single-period binomial HJM, 584–85 two-period binomial HJM, 606 Multiperiod binomial HJM model, 607 Multiperiod binomial options pricing model, 403–9 binomial coefficients and pseudo-probabilities in, 403 and BSM model, 407–9 features of, 399 general form of, 405–7

729

no-arbitrage principle for, 383 stock prices in, 393–94 and two-period binomial options pricing model, 403–5 Multiplicative binomial model, 377–78 Mutual funds, 120 n-factorial, 403, 686 N-period binomial options pricing model, 405–9 Naked short selling, 77 Naked trading strategies, 314–16 NASDAQ, see National Association of Securities Dealers Automated Quotations National Association of Securities Dealers Automated Quotations (NASDAQ), 61, 62 National Commodity and Derivatives Exchange Limited, 87 National Futures Association (NFA), 81, 160, 197 National Multi Commodity Exchange of India Ltd., 86–87 National Securities Clearing Corporation (NSCC), 60 Natural logarithms, 685–86 NAV (net asset value), 120 Necessary conditions, for no arbitrage, 570–71, 586–87, 597–98 Net asset value (NAV), 120 Net present value, risk premiumand, 386 Netherlands, 59, 73, 204, 291, 305 New York Mercantile Exchange (NYMEX), 84, 158 New York Stock Exchange (NYSE), 60 NFA, see National Futures Association No-arbitrage principle, 117 in binomial options pricing model, 376–77 and index arbitrage, 123 proof of, 384–88 No-dividend stocks, American options on, 358 Noncommercial traders, futures, 205 Nondiversifiable risk, 14 Noneconomic terms, in Master Agreements, 542 Nonfinancial companies, risk management for, 17 Nonlinear statistical model, BSM as, 479 Nonstandard derivatives, 94 Normal backwardation, 248, 249 Normal contango, 247, 248

730

SUBJECT INDEX

Normal distribution, 688–90 Notes commodity-indexed, 134 equity-linked, 134 exchange-traded, 168 Treasury, 43, 44, 530 Nothing comes from nothing principle, 117–18 Notional principal, in interest rate swaps, 136 Notional variables, 7 NSCC (National Securities Clearing Corporation), 60 Numbers, reporting, 26 Numerical methods, of pricing options, 369, 371, 448 NYMEX, see New York Mercantile Exchange NYSE, see New York Stock Exchange OCC, see Options Clearing Corporation Off-the-run securities, 40 Offer price, see Ask price Offset, in swap closing, 544 On-the-run securities, 40 One-by-two call spread, 325 One-factor models, 584 One price, law of, see Law of one price 1:2 hedge, 320 OPEC, see Organization of the Petroleum Exporting Countries Open-end companies, 119–20 Open interest, 87–88, 165 Open repos, 42 Opening call period, for futures, 165 Opening high price, 165 Opening low price, 165 Opening price (opening range), 165 Operational risk, 13, 661, 668, 678 Optimal hedge ratio (h), 273–75, 280 Option class, 301 Option contracts, see Options Options (option contracts), 92–109, 288–300; see also specific types, e.g.: American options decay of, 313 with dividends, 409–10, 444–46 exchange-traded, 105–6, 289–90 features of, 296–300 order placement strategies, 107–8 outright purchases of, 314 price quotes for, 301–4 quarterly cycles of, 297 terminology for, 93–94

Options Clearing Corporation (OCC), 105, 293, 299, 300 Options Industry Council, 319 Options market history of, 290–96 longs and shorts in, 107 regulation and manipulation of, 305–6 traders in, 311–12 Options on futures (futures options), 160, 296 Options on spot (spot options), 160, 295 Options prices, 336–63; see also specific pricing models, e.g.: Binomial options pricing model COP data, 313 and early exercise of American options, 357–62 history of options pricing, 369–71 and market manipulation, 439–40 put–call parity, 337–42 quotes for, 301–4 restrictions on, 346–56 Options series, 301 Options trading, 300–1, 310–32 combination strategies, 329–31 COP data for, 313 early exercise of American options, 357–62 hedged strategies, 317–20 history of, 312–13 naked strategies, 314–16 profit diagrams for, 312–13 spread strategies, 320–29 Optiver Holding, 204–5 Order placement for futures, 83–84 for options, 105 for stock, 59, 60 Organization of the Petroleum Exporting Countries (OPEC), 194 OTC Bulletin Board (OTCBB), 61 OTC markets, see Over-the-counter (OTC) markets OTC transactions, stock, 55 Out-of-the-money (term), 95, 101, 305 Output price risk, 89 Over-the-counter (OTC) markets, 3 forwards, 78–80 options, 289–90 stock, 55 trading securities on, 61 Overnight repos, 42 Overwrite, option, 317, 318 Owners, see Buyers

Pacific Exchange, 295 Palm Inc., 337, 343 Par bond yield, 506, 552 Par swaps, 145, 547 Par value, see Principal value Partial differential equation proof, of BSM model, 429, 458–59 Payer swaptions, 560 Payment dates of floorlets, 639 of forward rate agreements, 523 of interest rate caplets, 629 Payoff diagrams for bonds indexed to gold, 133 for call options, 95–97 for caps and caplets, 563 for cost-of-carry model, 212, 213 for put options, 101 put–call parity on, 337–41 Payoffs for caplets, 577–79, 627–28 for floors and floorlets, 563–64 for futures with early closes, 181 at liquidation, 132 for long vs. short forwards, 77–78 for zero-coupon bonds with forward rate, 517 PEPS (premium equity participating securities), 134 Percentage dividend rates, 409 Perfect hedges, 264, 268 Perfect markets, 95 Perfect substitutes hypothesis, 504 Performance margin, see Initial margin Philadelphia Stock Exchange, 295 Physical delivery, forwards and futures, 74, 76, 83, 174 Pink quote, 61 Plain vanilla (standard) derivatives, 94 Plain vanilla swaps, 135, 543 currency, 140–41, 145 interest rate, 94, 136–38, 544–49 Policy Analysis Market, 193–95 Ponzi schemes, 201 Portfolio risk, 14 Portfolio risk management, 14 Portfolio theory, 8 Portfolios delta- and gamma-neutral, 484–85 delta and vega-neutral portfolios, 485–86 delta-neutral, 483, 643, 644 duration-neutral, 512 futures, 182

SUBJECT INDEX

hedging of, 469–70 top-down construction of, 14 Position limits, 163, 299, 528 Position size, futures, 162 Position traders (trend followers), 57 Positive homogeneity, of risk measures, 667 Preferred habitat theory, 504 Premium option, 93, 94, 290 Premium equity participating securities (PEPS), 134 Present value, 34–35 Price bounds for American call options, 99–100 for American put options, 103–4 for forwards with market imperfections, 249–50 Price bubbles, 305 and cost-of-carry model, 215 and options pricing, 375, 424, 425 Price-contingent orders, 108 Price discovery, 156, 194 Price limit, daily, 162 Price quotes futures, 162, 164–66 options, 301–4 Price relatives, stock, 441 Price restrictions, options, 346–56 Price risks commodity, 15 input, 88–89 output, 89 Price-weighted average, 121 Price-weighted index (PWI), 121 Pricing and hedging argument, 428–29 Pricing models; see also specific models, e.g.: Black–Scholes–Merton model arbitrage in, 113 for currency swaps, 143–44 for derivatives, 12 Primary dealers, Treasuries, 39 Primary market, stock, 54 Principal, 27 Principal value, 7 of coupon bonds, 43 of zero-coupon bond, 32 Private placement, 134 Probabilities, see Actual probabilities; Pseudo-probabilities Probability distribution, for losses, 660–62 Procter & Gamble Corporation (P&G) interest rate swaps with, 539 risk management in, 16–17

Production assets, cost-of-carry model for, 214 Profit diagrams for call options, 96–98, 101, 103 for long vs. short forwards, 77–78 for options price restrictions, 346–48 for options trading, 312–13 Profit margin, 66 Profits; see also Arbitrage profits of hedged vs. unhedged firms, 263 and rate of return, 24–25 and zero-profit point, 313 Promised rate of return, 25 Protective put, 318, 319 Pseudo-probabilities, 379; see also Martingales actual and, 385, 433–34, 457–58, 576, 638 and binomial HJM model, 576 and binomial options pricing model, 403, 408 and BSM model, 408, 433–34, 457–58 computing, with Microsoft Excel, 413 and HJM libor model, 638 and no-arbitrage principle, 384 risk-neutral valuation with, 381, 582 and risk premium, 385–88 Put and Call Brokers and Dealers Association, 292 Put options (puts), 100–4; see also Put–call parity American, 104, 344–46, 348, 349, 352, 353 arbitrage opportunities with, 354–55 in binomial options pricing model, 373, 374 binomial pricing model for, 373, 374 BSM model for, 431–2 cash-secured, 316 on commodity futures, 296 covered, 319 defined, 93 delta and gamma for, 437–38 delta hedging errors with, 465 digital, 448 European, 348, 352–3, 374, 415–16, 431–32, 439 exercising, 93 on interest rates, see Floorlets intrinsic and time values of, 103 long, 298, 315, 316, 318, 319, 339, 344–45 married, 319 payoff and profit diagrams for, 100–2

731

price bounds for, 104, 348, 349, 352 pricing, with Microsoft Excel, 414–16 protective, 319 short, 315, 316, 318, 319, 345–46 and strike price, 352 Put spreads, 313, 321, 322 Put-to-call strategies, see Straddles Putable swaps, 550 Put–call parity, 337–46 and BSM model for pricing puts, 432–3 for European options, 338–42 and market imperfections, 342–46 on payoff diagrams, 337–39 pricing put options with, 401 and single-period binomial options pricing model, 374 PWI (price-weighted index), 121 Quality specifications, in futures trading, 163 Quarterly cycles, options, 297 Quarterly options (quarterlies), 297 Random walk processes, 452 Rate-capped swaps, 549 Rate of return, 24–25; see also Yield annualized, 25, 45, 48 promised vs. realized, 25 Ratio log, 633n2 Ratio spread, 324–26, 327n5 Real assets, 7 Real options, 672–73 Realized rate of return, 25 Receiver swaptions, 560 Reduced-form models, 371, 669, 673–76 Reference rate, FRA, 522 Regulation(s) and financial innovation, 7, 8 of futures market, 196–201 of options market, 305–6 of stock exchanges and OTC markets, 56 Regulation ATS, 61 Regulation Q, 8 Regulatory arbitrage, 342 Reinsurance, 327, 328–29 Repo rate, 41 Repurchase agreement (repo) market, 41–42 Resale market, US Treasury, 40 Return-to-maturity expectations (RME) hypothesis, 505

732

SUBJECT INDEX

Reuters/Jeffries CRB Index (RJ/CRB), 167 Reverse cash-and-carry, 217 Reverse repo market, 41–42 Reverse stock splits, 300 Reversing trades, 174 Rho, 436, 439 Rho hedging, 439 Rhône-Poulenc, 130 RICI (Rogers International Commodity index), 167 Risk; see also Credit risk basis, 264–7 classification of, 12–14 corporate, 14–15 currency, 15 diversifiable, 14 financial, 4 Herstatt, 669 hurricane, 327 input price, 88–89 interest rate, 15, 562, 641 legal, 13, 678 liquidity, 13, 660, 668, 677–78 market, 13, 660, 668–69 measures of, 667–68 model, 679–80 nondiversifiable, 14 nonhedged, 16 operational, 13, 661, 668, 678 output price, 89 portfolio, 14 price, 15, 88–89 settlement, 13, 669 tail, 329 Risk-adjusted interest rate, 382–83 Risk management, 12–19, 655–81 and Basel Committee’s four risks, 668–78 and classification of risk, 12–14 corporate financial, 14–17, 658–60 future of traded derivatives and, 679–80 with futures, 83 interest rate, 507–13 loss distribution in, 660–62 perspectives on, 17–18 portfolio, 14 scenario analysis in, 666–68 VaR in, 662–66 Risk Management Guidelines for Derivatives (Basel Committee on Banking), 13, 668 Risk-minimization hedging, 268–72

Risk-neutral probabilities, 379; see also Pseudo-probabilities Risk-neutral valuation in binomial HJM model, 581–83 and binomial options pricing model, 381–82, 400–1 and BSM model, 433, 456 of caplets, 602–4 and HJM libor model, 637–38 of zero-coupon bonds, 574, 600–1 Risk premiums and BSM model, 457–58 and cost-of-carry model, 246–47 and expectations hypothesis, 504 and pseudo-/actual probabilities, 385–87 of zero-coupon bonds, 576 Riskless securities, 568 Risky securities, 568 RJ/CRB (Reuters/Jeffries CRB Index), 167 RME (return-to-maturity expectations) hypothesis, 505 Robust assumptions, 679 Rogers International Commodity index (RICI), 167 Roller-coaster swaps, 550 Rolling, of futures indexes, 167 Round lot, 60 Round-trip commissions, for trading futures, 84 Rounding, conventions for, 26 Royal Exchange of London, 157 Sallie Mae (Student Loan Marketing Association), 540 Salomon Brothers, Inc., 135, 203–4, 311, 539, 678 Saving, role of derivatives in, 5 Savings and loans banks, 135, 498 Scalpers, 57 Scenario analysis, 666–68 Sealed bid auctions, 38 Sealed bids, 38 Seasonal swaps, 550 SEC, see Securities and Exchange Commission Secondary markets, 40, 54 Secured loans, 67 Securities; see also Derivatives; Treasury securities (Treasuries) asset-backed, 498 automated trading of, 58 defined, 7

fixed-income, 23; see also Bonds gilt, 143 off- vs. on-the-run, 40 premium equity participating, 134 riskless vs. risky, 568 Securities and Exchange Commission (SEC), 55 on calls against employee stock options, 318–19 on options, 290 regulation of options by, 200 regulation of stock exchanges by, 55 Securities markets, 54–57 Securitization, 38 Security futures products, 198 Self-financing trading strategy, 398 Self-regulatory organizations (SROs), 200 Sellers option, see Writers, option swap, 545 Selling hedges, 89 Selling the spread, 187 Semiannual bond equivalent basis, 548 Semistrong-form efficiency, 119 Settlement cash, 74, 76, 175 daily, 84, 176–80 of exchange-traded call options, 105–6 of exchange-traded futures, 84 final, 528 futures trading at, 163 of securities trades, 59, 60–61 Settlement date, FRA, 522, 523 Settlement price, futures, 166 Settlement risk, 13, 669 Shanghai Futures Exchange, 86 Shareholder/debt holder conflict, 671 Shareholder’s equity, 15, 659 Shipping certificates, for futures, 174 Short calls in covered trading strategies, 317–20 in 1:2 hedge, 320 profit diagrams for, 315, 316 put–call parity for American, 344–45 Short hedges, 89, 266 Short positions (shorts), 74 in covered trading strategies, 126, 318, 319 on options, 93, 107 payoffs for forwards with, 77–78 in stock, 314 Short puts in covered trading strategies, 318, 319

SUBJECT INDEX

profit diagrams for, 315 put–call parity for, 345–6 Short selling, 64–66 margin accounts for, 67, 68 naked, 77 proceeds from, 251n4 rate of return for, 25n2 in repo market, 41 Short squeeze, 125–26, 203–4 Short straddle, 331 Silver futures, manipulation of, 202–3 Simple forward rate, 613, 624–25, 631 Simple interest rates, 27 Single-period binomial HJM model, 570–85 and actual vs. pseudo-probabilities, 576 arbitrage-free conditions for, 570–73 caplet pricing with, 576–80 equal pseudo-probability condition for, 575–76 floorlet valuation with, 582–83 hedge ratio in, 581 and martingales, 573–75 multiple factors in, 584–85 risk-neutral valuation with, 581–82 and zero-coupon bond prices, 573–75 Single-period binomial options pricing model, 376–83 and actual vs. pseudo-probabilities, 382–83 arbitrage-free trees in, 378 binomial trees in, 377–78 for European calls, 372–75 hedge ratio for, 380–81 martingales in, 378–79 multiperiod vs., 399 no-arbitrage principle for, 376–77, 383 risk-neutral valuation in, 381–82 stock prices in, 378–79, 393–94 Southwest Airlines, 260–61 S&P 500, see Standard & Poor’s 500 Index S&P GSCI indexes, 167 Speculation, 5, 10–11 with caps, 561–62 in futures market, 195–96 Speculators, 57 Spin-offs, 343 Split ratio, 299 Spot commodity positions, 269–70 Spot options, 160, 295 Spot prices, 76, 183, 186, 246–49

Spot rate, 500, 502, 518, 565, 570, 595 Spread orders, 108 Spread trading, 120–21 futures, 185–88 options, 320–29 Spreads, 18 bear, 321–23 bid/ask, 56, 251 bull, 321–23, 324, 326 butterfly, 326, 326 buying, 187 call, 321, 323, 324, 326, 327 call backspread, 327n5 call ratio, 327n5 CDS, 148 condor, 325, 326 credit, 321 debit, 321 diagonal, 321 horizontal (time), 185, 321 intercommodity, 185, 188 intracommodity, 185, 187 one-by-two call, 325 put, 313, 321, 322 ratio, 324–26, 327n5 selling, 187 Treasury–Eurodollar, 48, 500, 554 vertical, 321–24 SROs (self-regulatory organizations), 200 Standard & Poor’s 500 Index (S&P 500), 63, 121, 123, 160 Standard normal random variable, 426–27 Standardization, futures market, 193–94 Statistical models, 474–75, 636 Stochastic volatility option models, 472–73 Stock, 59–70 defined, 7 dividend-paying, 234 dividends of, 62–64, 232–43 forwards on, 232–43 margin accounts for trading, 66–68 short selling, 64–66 transactions with, 58–59 Stock index funds, 235 Stock indexes, 63, 121–23 Stock prices in binomial options pricing model, 378–79, 393–94 binomial tree for, 395–96 in BSM model, 432–33

733

with continuous vs. discrete trading, 427 lognormal assumption for, 426, 451–59 and price of call option, 350 Stock privileges, 329 Stock splits, 299–300 Stock trading alternative systems for, 61–62 on exchanges, 60–61 and margin, 66–68 at NYSE, 59–62 on OTC securities market, 61 short selling in, 64–66 Storage costs, 243–45 Straddles, 330–31 Straight cancel orders, 108 Strangles, 330–31 Stress testing (scenario analysis), 666–68 Strike prices, 93, 297–98, 350, 352 STRIPS (separate trading of registered interests and principal of securities), 44–45 Strong-form efficiency, 119 Structural models, 371, 669–73 Student Loan Marketing Association (Sallie Mae), 540 Subadditivity, of risk measures, 667 Sufficient conditions, for no arbitrage, 572–73, 587, 597–99 Superglue argument, 346, 352–54 Supply chains, commodities in, 86 Swap books, managing, 541 Swap buyers, 545 Swap curve, 552, 554 Swap dealers, 205–6, 541 Swap execution facilities, 199 Swap facilitators (swap banks), 136, 541 Swap issuers, 542 Swap market, 540 Swap rate, 547, 553 Swap sellers, 545 Swaps, 135–48; see also Interest rate swaps; Plain vanilla swaps accreting, 550 amortized, 549 application and uses of, 135–36 asset-backed securities credit default, 148 callable, 550 cancellable, 550 commodity, 146–47 constant maturity, 550

734

SUBJECT INDEX

credit default, 147–48, 676 currency, 139–47 development of, 8 equity, 146–47 exotic, 94 forex, 138–39 forward, 550 par, 145, 547 putable, 550 rate-capped, 549 roller-coaster, 550 seasonal, 550 terminology for, 135 yield, 549 Swaptions, 560, 611–12, 647 Synthetic calls, 372, 397–400 Synthetic caplets, 578–79, 601–2 Synthetic futures contracts, 121 Synthetic indexes, 236–39 Synthetic interest rate swaps, 550–52 Synthetic options, 372, 376, 397–400, 429 T-bills (Treasury bills), 43–47 T-bonds (Treasury bonds), 47–47, 524 T-notes (Treasury notes), 43, 49, 530 Tail risk, 329 TARP, see Troubled Asset Relief Program TAS (Trading at Settlement) contracts, 163 Tax arbitrage, 342 Tax shields, 263 Taxes and financial innovation, 7, 8 and hedging strategies, 262–64 Taylor series, 482–87, 691–92 Technical analysis, 118 TED spread, see Treasury–Eurodollar spread Tenor, of swaps, 135 Term repos, 42 Term structure, of interest rates, 371, 502, 567–68, 594, 647 Termination date, FRA, 522 Theoretical models, 475–75, 636 Theta, 436 Third market makers, 59 35 percent rule, 203 3Com, 343 Tick size, 162, 297, 527 Ticker symbols, 162, 190 Time of forwards, 73, 74–75

measuring, 26 moving cash flows across, 31–37 Time-contingent orders, 108 Time-of-day orders, 108 Time spread, 185, 321 Time value debt, 671–72 of options, 99, 103 TIPS (Treasury inflation protected security), 43 Top-down portfolio construction, 14 Top straddle, 331 Total return indexes, 236, 238 Traders commercial, 205 commodity index, 205 day, 57 noncommercial, 205 in options market, 311–12 position, 57 risk management by, 17 stock market, 56–57 Trades bearish, 65, 93 block, 60 bullish, 65, 93 clearing of, 59, 60–61, 84, 105–6 reversing, 174 Trading; see also Futures trading; Options trading; Stock trading alternative systems for, 61–62 arbitrage in, 119–23 automated, 58, 295 covered strategies, 314, 317–20 discrete, 427 high-frequency, 425 liquidation-only, 203 naked strategies, 314–16 self-financing strategy, 398, 399 spread, 120–21, 185–88, 320–29 Trading at Settlement (TAS) contracts, 204 Trading futures, 174–76 Trading hours, 162, 528 Trading month, 162 Trading rooms, 79 Trading (ticker) symbols, 162 Trading unit, 162 Trading volume, 87–88 Transaction costs, 7, 9 Translation invariance, of risk measures, 667–68 Treasury bill futures, 520, 529–30 Treasury bills (T-bills), 43–47

Treasury bonds (T-bonds), 43–47, 552 Treasury futures, 529–30 Treasury inflation protected security (TIPS), 43 Treasury notes (T-notes), 43–47, 552 Treasury securities (Treasuries), 37–47 characteristics of, 37–38 and federal debt auction markets, 38–39 futures manipulation of, 203–4 investment methods for, 40–43 Treasury–Eurodollar spread, 48 types of, 43–47 yield and swap curve for, 503, 552, 554 Treasury–Eurodollar (TED) spread, 48, 500, 554 Trend followers, 57 Troubled Asset Relief Program (TARP), 289, 368, 666 Tulip bulb bubble, 305 Two-factor models, 584 Two-period binomial HJM model, 597–606 arbitrage-free conditions for, 597–99 caplet pricing with, 601–4 for Eurodollar futures, 608–10 floorlet pricing with, 605–6 for forward rate agreements, 607–8 with multiple factors, 606 for swaptions, 611–12 and zero-coupon bond prices/martingales, 599–601 Two-period binomial options pricing model, 396–402 and backward induction, 396–97 and multiperiod binomial options pricing model, 403–5 one-step valuation with, 401–2 and risk-neutral pricing, 400–1 and synthetic option construction, 397–400 Two-period binomial tree, 598–99 UITs (unit investment trusts), 120 Unbiased expectations (UE) hypothesis, 505 Uncertainty, lognormal assumption for stock prices and, 452–53 Uncovered trading strategies, for options, 314, 316 Under-margined accounts, 179 Underlying, for derivatives, 7, 527 Unit investment trusts (UITs), 120

SUBJECT INDEX

United Stock Exchange (USE), 87 US Department of Agriculture (USDA), 197 US Department of Defense, 193–94 US dollar, 6, 138 US Mint, 116 US Treasury, 38–39, 43–47, 368 Up factors, 377, 378, 408 USDA (US Department of Agriculture), 197 USE (United Stock Exchange), 87 Valuation; see also Risk-neutral valuation with binomial options pricing model, 401–2 of derivatives, 12 of interest rate swaps, 544–49 of plain vanilla currency swaps, 145 of swaptions, 611–12 Value-at-risk (VaR), 662–66 Value Line Index, 160 Value-weighted index (VWI), 122–23 VaR (value-at-risk), 662–66 Variance, historical volatility and, 443 Variation margin, 176 Vega, 436 Vega hedging, 439, 470–73, 480, 485–86, 641 Vertical spread, 321–24 VIX (CBOE Volatility index), 294, 480 Volatility; see also Vega annualized, 443 average forward rate, 632–36 of futures prices, 194 historical, 441–44, 632–36 implied, 444, 477–80, 632, 636 and value-at-risk, 664

735

Volatility smile, 478 Volume-weighted average price (VWAP), 205 VWI (value-weighted index), 122–23

for Treasury securities, 503 zero-coupon, 501–2 Yield swaps, 549 Yield to maturity, see Yield

Warehouse receipts, for futures, 174 Warehousing swaps, 541 Weak-form efficiency, 118, 119 Well-functioning market assumption, 215, 375, 424, 566, 623 “When, as, and if issued” trading, 40 When-issued market, 40 Wild card option, for T-bond futures, 530 World Bank, 539–40 Writers, option, 93 margin requirements for, 298–99 payoffs for, 96 profits for, 97–98 uncovered, 316

Zero-cost collars, 323, 324, 564 Zero-coupon bond prices, 32 and binomial HJM model, 573–75, 594–95, 599–601 and forward rate agreements, 631–32 and forward rates, 517–19 and martingales, 573–75, 599–601 single-period, 569–70 from swap rates, 553 term structure of, 567–68, 594 two-period, 594, 595 Zero-coupon bonds (zeroes), 7 compounding and discounting cash flows with, 32–33 and continuously compounded rate, 36 coupon bonds vs., 43 in currency swap valuation, 143 financial innovation with, 8 moving cash flows across time with, 34–37 payoff table for, 517 risk-neutral valuation of, 600, 600–1 risk premiums of, 576 in United Kingdom and US, 143 yields on, 499–501 Zero-coupon yield curve (zero curve), 501–2, 516 Zero net supply markets, 10, 74 Zero-profit point, 313 Zero-sum game, 10, 75, 77 Zhengzhou Commodity Exchange, 86

Yield, 499–507 banker’s discount, 45–47 bond equivalent, 45–47 computing, 506 convenience, 244–47 dividend, 63–64, 235–36, 237–39, 344 and expectations hypothesis, 504–5 and forward rate, 515, 516 par bond, 506, 552 and yield curves, 500–3, 506–7 and zero-coupon bonds, 499–500 Yield auctions, 38 Yield curves, 500–3, 506–7 and forward rates, 516 and swap curves, 552, 554